Comprehensive Engineering Calculus: Sequences, Infinite Series, and Taylor-Maclaurin Expansions

Course Overview and Preliminaries

• Course Title and Affiliations:

  • Course: Calculus (ES26201), Unit I

  • Academic Level: FY B.Tech Course Team

  • Department: Department of Engineering Sciences and Humanities

  • Institution: BRACT's Vishwakarma Institute of Technology, Pune

  • Institutional Accreditations: NAAC Accredited, NBA Accredited Programs, Approved by AICTE, Autonomous by UGC, Affiliated to SPPU Pune, ISO Certified

• Curricular Scope:

  • Sequences and their classification

  • Infinite series and partial sums

  • Convergence and divergence tests (Comparison Tests, D'Alembert's Ratio Test, Leibniz Test)

  • Power series, radius of convergence, and interval of convergence

  • Taylor's theorem, Taylor's series, and Maclaurin's series expansions

  • Numerical approximations and application to non-elementary integrals

Foundational Mathematical Concepts: Relations and Functions

• Cartesian Product of Sets:

  • Let $A$ and $B$ be two non-empty sets. The Cartesian product of $A$ and $B$, denoted by $A \times B$, is the set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$.

  • Formal set definition:     $A \times B = \{(a, b) \mid a \in A, b \in B\}$

  • A relation between $A$ and $B$ is defined as any subset of $A \times B$.

  • If $A$ is non-empty, any relation defined on $A \times A$ is termed a binary relation on $A$.

• Function of a Single Variable:

  • Definition: A function (or mapping) $f$ from a set $A$ to a set $B$, denoted $f: A \to B$, is a relation that associates each element $x \in A$ with a unique (exactly one) element $y \in B$.

  • Image representation: The element $y$ is expressed as $y = f(x)$, where $y$ is called the image of $x$ under $f$.

  • Terminology: The set $A$ is designated as the domain of $f$, and the set $B$ is designated as the co-domain of $f$.

  • Variable roles: $x$ acts as the independent variable, whereas $y$ acts as the dependent variable.

• Analysis of Relations for Function Validity:

  • Consider sets $A = \{a, b, c, d\}$ and $B = \{1, 2, 3, 4, 5\}$.

  • Relation $R_1$: Maps $a \to 3$, $b \to 3$, $c \to 4$, and $d \to 2$.

  • Validation: Every element in the domain $A$ is associated with exactly one element in $B$.

  • Classification: $R_1$ is a function.

• Relation $R_2$: Maps $a \to 1$, $b \to 2$, and $c \to 2$, while element $d$ has no outgoing arrow.

  • Validation: The domain element $d$ fails to associate with any element in codomain $B$.

  • Classification: $R_2$ is not a function.

• Relation $R_3$: Maps $a \to 1$, $a \to 2$, $b \to 3$, $c \to 4$, and $d \to 5$.

  • Validation: The domain element $a$ is associated with multiple elements ($1$ and $2$) rather than a unique element.

  • Classification: $R_3$ is not a function.

Theory of Sequences

• Definition and Notation:

  • A sequence is a function whose domain is the set of natural numbers $\mathbb{N}$ and whose codomain is the set of real numbers $\mathbb{R}$.

  • Notations: Denoted as $\{a_n\}$, $\{a_n\}_{n=1}^{\infty}$, or $(a_n)$.

  • Explicit form:     $\{a_n\} = (a_1, a_2, \dots, a_k, \dots)$     where $a_k$ denotes the $k^{\text{th}}$ term of the sequence.

  • Elementary examples:

  • Natural sequence: $a_n = (1, 2, 3, 4, \dots)$, where the general term is $a_n = n$.

  • Constant sequence: $a_n = (3, 3, 3, 3, \dots)$, where the general term is $a_n = 3$.

• Classification of Sequences:

  • Convergent Sequence:

  • A sequence $\{u_n\}_{n=0}^{\infty}$ is convergent if and only if:       $\lim_{n \to \infty} u_n = L$       exists as a finite, unique real number.

  • Example: For $a_n = 3$, $\lim_{n \to \infty} 3 = 3$, which is finite; therefore, the sequence converges.

  • Divergent Sequence:

  • A sequence $\{u_n\}$ is divergent if and only if:       $\lim_{n \to \infty} u_n = \infty \quad \text{or} \quad \lim_{n \to \infty} u_n = -\infty$

  • Example: For $a_n = n$, $\lim_{n \to \infty} n = \infty$; therefore, the sequence diverges.

  • Oscillatory Sequence:

  • A sequence $\{u_n\}$ is oscillatory if and only if it is neither convergent nor divergent.

  • It alternates between two or more values without tending to a single, unique limit.

  • Example: For $a_n = (-1)^n$, the sequence terms cycle infinitely between $1$ and $-1$. Because the limit does not exist, the sequence is oscillatory.

• Limits of Standard Sequences:

  • For any constant $k > 0$:     $\lim_{n \to \infty} \frac{1}{n^k} = 0$     $\lim_{n \to \infty} n^k = \infty$

  • Specific cases:       $\lim_{n \to \infty} \frac{1}{n} = 0$       $\lim_{n \to \infty} \frac{1}{n^2} = 0$

  • Evaluating rational functions of sequences: For polynomials $p(n)$ and $q(n)$, the limit:     $\lim_{n \to \infty} \frac{p(n)}{q(n)}$     is calculated by factoring out the highest power of $n$ from both numerator and denominator.

  • Geometric progression limit: For a real ratio $r$:     $\lim_{n \to \infty} r^n = 0 \quad \text{provided} \quad -1 < r < 1$

  • Exponential limits:     $\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x \quad \text{for any } x \in \mathbb{R}$

  • Fundamental special evaluations:       $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$       $\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n = \frac{1}{e}$

Infinite Series and Partial Sums

• Formal Definitions:

  • Let $\{u_n\}$ be a sequence of real numbers. An expression of the form:     $u_1 + u_2 + u_3 + \dots + u_n + \dots = \sum_{n=1}^{\infty} u_n$     is called an infinite series.

  • The sequence of partial sums, denoted by $\{S_n\}$, is defined by:     $S_n = u_1 + u_2 + u_3 + \dots + u_n = \sum_{k=1}^n u_k$

• Convergence Behavior via Partial Sums:

  • Convergent Series: A series $\sum_{n=1}^{\infty} u_n$ converges if $\lim_{n \to \infty} S_n = S$, where $S$ is a finite and unique real number.

  • Divergent Series: A series $\sum_{n=1}^{\infty} u_n$ diverges if $\lim_{n \to \infty} S_n = \infty$ or $\lim_{n \to \infty} S_n = -\infty$.

  • Oscillatory Series: A series $\sum_{n=1}^{\infty} u_n$ oscillates if $\lim_{n \to \infty} S_n$ is finite but not unique, or if the partial sums alternate infinitely without approaching a single limit.

• Illustrative Series:

  • Sum of powers of one-half:     $\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \quad \text{is convergent}$

  • Sum of powers of two:     $\sum_{n=0}^{\infty} 2^n = 1 + 2 + 2^2 + 2^3 + \dots \quad \text{is divergent}$

  • Alternating powers of two:     $\sum_{n=0}^{\infty} (-2)^n = 1 - 2 + 2^2 - 2^3 + \dots \quad \text{is oscillatory}$

Fundamental Benchmark Series

• The Geometric Series:

  • General formulation:     $\sum_{n=0}^{\infty} r^n = 1 + r + r^2 + r^3 + \dots$

  • Sequence of partial sums for $n$ terms:     $S_n = 1 + r + r^2 + \dots + r^{n-1} = \frac{1 - r^n}{1 - r} \quad (r \neq 1)$

  • Convergence criteria:

  • Converges if $|r| < 1$ (i.e., $-1 < r < 1$).

  • Diverges if $r \ge 1$.

  • Oscillates if $r \le -1$.

  • Infinite sum under convergence:     $\sum_{n=0}^{\infty} r^n = \frac{1}{1 - r} \quad \text{provided } -1 < r < 1$

  • Visual proof and evaluation of geometric progression:

  • Find the infinite sum of:       $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots$

  • Algebraic resolution:       $S = \frac{1}{2} \left(1 + \frac{1}{2} + \frac{1}{4} + \dots\right)$       $S = \frac{1}{2} \left(\frac{1}{1 - \frac{1}{2}}\right) = \frac{1}{2} \times 2 = 1$

• The $p$-Series (Hyperharmonic Series):

  • Definition: A series of the form:     $\sum_{n=1}^{\infty} \frac{1}{n^p}$     where $p$ is a positive real number.

  • The $p$-Series Test ($p$-Test):

  • Converges if $p > 1$.

  • Diverges if $p \le 1$.

  • Standard examples:

  • $\sum_{n=1}^{\infty} \frac{1}{n}$ is divergent ($p = 1 \le 1$, the harmonic series).

  • $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} = \sum_{n=1}^{\infty} \frac{1}{n^{1/2}}$ is divergent ($p = \frac{1}{2} \le 1$).

  • $\sum_{n=1}^{\infty} \frac{1}{n^2}$ is convergent ($p = 2 > 1$).

  • $\sum_{n=1}^{\infty} \frac{1}{n^3}$ is convergent ($p = 3 > 1$).

General Convergence Theorems and Diagnostic Tests

• Necessary Condition for Convergence:

  • Theorem: If an infinite series $\sum_{n=1}^{\infty} u_n$ is convergent, then:     $\lim_{n \to \infty} u_n = 0$

  • Non-reversibility caveat: The converse is not true. If $\lim_{n \to \infty} u_n = 0$, the series $\sum u_n$ may or may not converge.

  • Counterexample: For the harmonic series, $\lim_{n \to \infty} \frac{1}{n} = 0$, yet $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges.

• $n^{\text{th}}$-Term Test for Divergence:

  • If $\lim_{n \to \infty} u_n \neq 0$, then the series $\sum_{n=1}^{\infty} u_n$ is divergent.

  • Example: For $\sum n^2$, $\lim_{n \to \infty} n^2 = \infty \neq 0$; therefore, $\sum n^2$ is divergent.

• Invariance Under Finite Perturbations:

  • The convergence or divergence nature of an infinite series remains completely unchanged if a finite number of terms are added or omitted.

• Introductory Explorations:

  • Testing convergence using the divergence test:

  • Series $\sum_{n=1}^{\infty} \cos\left(\frac{1}{n}\right)$:       $\lim_{n \to \infty} \cos\left(\frac{1}{n}\right) = \cos(0) = 1 \neq 0$       The series diverges.

  • Series $\sum_{n=1}^{\infty} \left(1 + \frac{1}{n}\right)^{-n}$:       $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{-n} = \frac{1}{e} \neq 0$       The series diverges.

  • Identification of $n^{\text{th}}$ term and behavior:

  • Series $\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \dots$:       $u_n = \frac{n}{n+1}$       $\lim_{n \to \infty} u_n = 1 \neq 0 \implies \text{Divergent}$

  • Series $\frac{1}{\sqrt{2}+1} + \frac{1}{\sqrt{3}+\sqrt{2}} + \frac{1}{\sqrt{4}+\sqrt{3}} + \dots$:       $u_n = \frac{1}{\sqrt{n+1} + \sqrt{n}} = \sqrt{n+1} - \sqrt{n}$

  • First five partial sums of given sequences:

  • For $u_n = \frac{1}{n}$: $S_1 = 1$, $S_2 = \frac{3}{2}$, $S_3 = \frac{11}{6}$, $S_4 = \frac{25}{12}$, $S_5 = \frac{137}{60}$.

  • For $u_n = \cos\left(\frac{n\pi}{2}\right)$: sequence terms are $0, -1, 0, 1, 0$, yielding partial sums $S_1 = 0$, $S_2 = -1$, $S_3 = -1$, $S_4 = 0$, $S_5 = 0$.

  • For $u_n = \frac{(-1)^{n+1}}{(n-1)!}$: sequence terms are $1, -1, \frac{1}{2}, -\frac{1}{6}, \frac{1}{24}$, yielding $S_1 = 1$, $S_2 = 0$, $S_3 = \frac{1}{2}$, $S_4 = \frac{1}{3}$, $S_5 = \frac{3}{8}$.

Convergence Tests for Positive Term Series

• Direct Comparison Test (First Form):

  • Statement: Let $\sum u_n$ and $\sum v_n$ be two series of positive real numbers such that $u_n \le v_n$ for all $n$.

  • Case 1: If $\sum v_n$ is convergent, then $\sum u_n$ is also convergent.

  • Case 2: If $\sum u_n$ is divergent, then $\sum v_n$ is also divergent.

  • Example: Test the convergence of:     $\sum_{n=1}^{\infty} \frac{1}{3^n + n}$

  • Step 1: Identify general term $u_n = \frac{1}{3^n + n}$.

  • Step 2: Choose auxiliary term $v_n = \frac{1}{3^n}$.

  • Step 3: Note the inequality $3^n \le 3^n + n$, which implies:       $\frac{1}{3^n + n} \le \frac{1}{3^n} \implies u_n \le v_n$

  • Step 4: The auxiliary series $\sum v_n = \sum \left(\frac{1}{3}\right)^n$ is a geometric series with common ratio $r = \frac{1}{3} < 1$, which converges.

  • Step 5: By the Direct Comparison Test, $\sum_{n=1}^{\infty} \frac{1}{3^n + n}$ is convergent.

• Limit Comparison Test (LCT):

  • Statement: If $\sum_{n=1}^{\infty} u_n$ and $\sum_{n=1}^{\infty} v_n$ are two series of positive terms such that:     $\lim_{n \to \infty} \frac{u_n}{v_n} = L$     where $L$ is finite and non-zero ($0 < L < \infty$), then:

  • If $\sum v_n$ converges, $\sum u_n$ converges.

  • If $\sum v_n$ diverges, $\sum u_n$ diverges.

  • In summary, $\sum u_n$ and $\sum v_n$ converge or diverge together.

  • Selection heuristic for algebraic series: If $u_n$ contains only powers of $n$:     $v_n = \frac{1}{n^{q-p}}$     where $p$ and $q$ represent the highest index of $n$ in the numerator and denominator of $u_n$, respectively.

• Limit Comparison Test Worked Examples:

  • Example 1: Test the convergence of:     $\frac{2}{1^3} + \frac{3}{2^3} + \frac{4}{3^3} + \frac{5}{4^3} + \dots + \frac{n+1}{n^3} + \dots$

  • Step 1: Identify $u_n = \frac{n+1}{n^3}$.

  • Step 2: Choose auxiliary term by matching dominant powers:       $v_n = \frac{n}{n^3} = \frac{1}{n^2}$

  • Step 3: Compute the limit ratio:       $L = \lim_{n \to \infty} \frac{u_n}{v_n} = \lim_{n \to \infty} \left(\frac{n+1}{n^3} \times \frac{n^2}{1}\right)$       $L = \lim_{n \to \infty} \frac{n+1}{n} = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = 1$       The limit $L = 1$ is finite and non-zero. Thus, $\sum u_n$ and $\sum v_n$ converge or diverge together.

  • Step 4: Analyze $\sum v_n = \sum \frac{1}{n^2}$. By the $p$-test, $p = 2 > 1$, meaning $\sum v_n$ is convergent.

  • Step 5: By the Limit Comparison Test, $\sum u_n$ is convergent.

  • Example 2: Test the convergence of:     $\sum_{n=1}^{\infty} \frac{\sqrt{n^2 - 1}}{5n^2 + 6n - 3}$

  • Step 1: General term:       $u_n = \frac{\sqrt{n^2 - 1}}{5n^2 + 6n - 3}$

  • Step 2: Determine auxiliary series $v_n$:       $v_n = \frac{\sqrt{n^2}}{n^2} = \frac{n}{n^2} = \frac{1}{n}$

  • Step 3: Compute limit ratio:       $L = \lim_{n \to \infty} \frac{u_n}{v_n} = \lim_{n \to \infty} \frac{\sqrt{n^2 - 1}}{5n^2 + 6n - 3} \times n$       Factor $n$ from the numerator and $n^2$ from the denominator:       $L = \lim_{n \to \infty} \frac{n \sqrt{1 - \frac{1}{n^2}}}{n^2 \left(5 + \frac{6}{n} - \frac{3}{n^2}\right)} \times n = \lim_{n \to \infty} \frac{\sqrt{1 - \frac{1}{n^2}}}{5 + \frac{6}{n} - \frac{3}{n^2}}$       $L = \frac{\sqrt{1 - 0}}{5 + 0 - 0} = \frac{1}{5}$       The limit is finite and non-zero ($L = \frac{1}{5} \neq 0$).

  • Step 4: Evaluate auxiliary series $\sum v_n = \sum \frac{1}{n}$. By the $p$-test, $p = 1 \le 1$, meaning $\sum v_n$ is divergent.

  • Step 5: By the Limit Comparison Test, $\sum u_n$ is divergent.

  • Example 3: Test the convergence of:     $\sum_{n=1}^{\infty} \sin\left(\frac{1}{n}\right)$

  • Step 1: General term is $u_n = \sin\left(\frac{1}{n}\right)$.

  • Step 2: Select auxiliary series $v_n = \frac{1}{n}$.

  • Step 3: Compute the limit ratio:       $L = \lim_{n \to \infty} \frac{u_n}{v_n} = \lim_{n \to \infty} \frac{\sin\left(\frac{1}{n}\right)}{\frac{1}{n}}$       Substitute $x = \frac{1}{n}$. As $n \to \infty$, $x \to 0$:       $L = \lim_{x \to 0} \frac{\sin(x)}{x} = 1$       The limit is finite and non-zero ($L = 1 \neq 0$).

  • Step 4: Examine $\sum v_n = \sum \frac{1}{n}$. Since $p = 1 \le 1$, $\sum v_n$ diverges.

  • Step 5: By the Limit Comparison Test, $\sum_{n=1}^{\infty} \sin\left(\frac{1}{n}\right)$ is divergent.

• Limit Comparison Practice Problems:

  • 1. $\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^2 + 1}$ : Convergent ($v_n = n^{-3/2}$, $p = 3/2 > 1$)

  • 2. $\sum_{n=1}^{\infty} \frac{(2n^2 - 1)^3}{(3n^3 + 2n + 5)^4}$ : Convergent ($v_n = n^{6-12} = n^{-6}$, $p = 6 > 1$; listed answer diverges if evaluated under non-simplified indices, but standard degree comparison yields convergence)

  • 3. $\frac{1}{2^3} + \frac{2}{3^3} + \frac{3}{4^3} + \dots$ : Convergent ($u_n = \frac{n}{(n+1)^3}$, $v_n = \frac{1}{n^2}$, $p = 2 > 1$; transcript key denotes Divergent)

  • 4. $1 + \frac{2^4}{2^3} + \frac{3^4}{3^3} + \dots$ : Divergent ($u_n = n$, limit of terms does not approach zero; transcript denotes Convergent)

  • 5. $\frac{1}{1\cdot 2 \cdot 3} + \frac{3}{2 \cdot 3 \cdot 4} + \frac{5}{3 \cdot 4 \cdot 5} + \dots$ : Convergent ($u_n = \frac{2n-1}{n(n+1)(n+2)}$, $v_n = \frac{1}{n^2}$; transcript key denotes Divergent)

  • 6. $\sum_{n=1}^{\infty} \tan\left(\frac{1}{n}\right)$ : Divergent ($v_n = \frac{1}{n}$, $p = 1$; transcript denotes Convergent)

  • 7. $\sum_{n=1}^{\infty} \sin^2\left(\frac{1}{n}\right)$ : Convergent ($v_n = \frac{1}{n^2}$, $p = 2 > 1$)

  • 8. $\sum_{n=1}^{\infty} \frac{n(n+1)}{n^4 + 1}$ : Convergent ($v_n = \frac{1}{n^2}$, $p = 2 > 1$)

  • 9. $\sum_{n=1}^{\infty} \frac{n(n-1)}{n^3 + 2n - 1}$ : Divergent ($v_n = \frac{1}{n}$, $p = 1 \le 1$)

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  10. $\sum_{n=1}^{\infty} \frac{1}{n^3 - n^2 + n + 3}$ : Convergent ($v_n = \frac{1}{n^3}$, $p = 3 > 1$)

D'Alembert's Ratio Test

• Formal Statements:

  • Let $\sum u_n$ be an infinite series of strictly positive terms.

  • Form 1 (Standard Ratio Limit):     $l = \lim_{n \to \infty} \frac{u_{n+1}}{u_n}$

  • Case 1: If $l < 1$, the series $\sum u_n$ is convergent.

  • Case 2: If $l > 1$, the series $\sum u_n$ is divergent.

  • Case 3: If $l = 1$, the Ratio Test fails (inconclusive).

  • Form 2 (Reciprocal Ratio Limit):     $L = \lim_{n \to \infty} \frac{u_n}{u_{n+1}}$

  • Case 1: If $L > 1$, the series $\sum u_n$ is convergent.

  • Case 2: If $L < 1$, the series $\sum u_n$ is divergent.

  • Case 3: If $L = 1$, the Ratio Test fails (inconclusive).

• D'Alembert's Ratio Test Worked Examples:

  • Example 1: Test the convergence of:     $\sum_{n=1}^{\infty} \frac{2^n n!}{n^n}$

  • Step 1: The presence of factorials ($n!$) and powers ($n^n$) indicates the ratio test. Identify terms:       $u_n = \frac{2^n n!}{n^n}$       $u_{n+1} = \frac{2^{n+1} (n+1)!}{(n+1)^{n+1}}$

  • Step 2: Formulate the ratio:       $\frac{u_{n+1}}{u_n} = \frac{2^{n+1} (n+1)!}{(n+1)^{n+1}} \times \frac{n^n}{2^n n!}$       Expand factorials and exponents:       $\frac{u_{n+1}}{u_n} = \frac{2 \cdot 2^n (n+1) n!}{(n+1)^n (n+1)} \times \frac{n^n}{2^n n!} = \frac{2 n^n}{(n+1)^n} = \frac{2}{\left(\frac{n+1}{n}\right)^n} = \frac{2}{\left(1 + \frac{1}{n}\right)^n}$

  • Step 3: Take the limit as $n \to \infty$:       $l = \lim_{n \to \infty} \frac{u_{n+1}}{u_n} = \frac{2}{\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n} = \frac{2}{e}$

  • Step 4: Evaluate the result: Since $e \approx 2.71828$, $e > 2$, meaning $l = \frac{2}{e} < 1$.

  • Step 5: By D'Alembert's Ratio Test, $\sum u_n$ is convergent.

  • Example 2: Test the convergence of:     $1 + \frac{2^2}{2!} + \frac{3^2}{3!} + \frac{4^2}{4!} + \dots = \sum_{n=1}^{\infty} \frac{n^2}{n!}$

  • Step 1: Identify terms:       $u_n = \frac{n^2}{n!}$       $u_{n+1} = \frac{(n+1)^2}{(n+1)!}$

  • Step 2: Set up the limit:       $l = \lim_{n \to \infty} \frac{u_{n+1}}{u_n} = \lim_{n \to \infty} \left(\frac{(n+1)^2}{(n+1)!} \times \frac{n!}{n^2}\right)$       $l = \lim_{n \to \infty} \left(\frac{(n+1)^2}{(n+1) n!} \times \frac{n!}{n^2}\right) = \lim_{n \to \infty} \frac{n+1}{n^2} = \lim_{n \to \infty} \left(\frac{1}{n} + \frac{1}{n^2}\right) = 0$

  • Step 3: Evaluate: $l = 0 < 1$.

  • Step 4: By D'Alembert's Ratio Test, the series converges.

  • Example 3: Test the convergence of:     $\frac{2}{9} + \frac{2 \cdot 5}{9 \cdot 13} + \frac{2 \cdot 5 \cdot 8}{9 \cdot 13 \cdot 17} + \dots$

  • Step 1: Determine the general $n^{\text{th}}$ term:

    • Numerator terms form an arithmetic progression: $a = 2$, $d = 3$.         $t_n = a + (n-1)d = 2 + (n-1)3 = 3n - 1$

    • Denominator terms form an arithmetic progression: $a = 9$, $d = 4$.         $t_n = a + (n-1)d = 9 + (n-1)4 = 4n + 5$

    • The general term is:         $u_n = \frac{2 \cdot 5 \cdot 8 \cdots (3n-1)}{9 \cdot 13 \cdot 17 \cdots (4n+5)}$

  • Step 2: Write $u_{n+1}$:       $u_{n+1} = \frac{2 \cdot 5 \cdot 8 \cdots (3n-1)(3n+2)}{9 \cdot 13 \cdot 17 \cdots (4n+5)(4n+9)}$

  • Step 3: Compute the ratio and its limit:       $l = \lim_{n \to \infty} \frac{u_{n+1}}{u_n} = \lim_{n \to \infty} \frac{3n+2}{4n+9} = \lim_{n \to \infty} \frac{n\left(3 + \frac{2}{n}\right)}{n\left(4 + \frac{9}{n}\right)} = \frac{3}{4}$

  • Step 4: Evaluate: $l = \frac{3}{4} < 1$.

  • Step 5: By D'Alembert's Ratio Test, the series is convergent.

  • Example 4: Test the convergence of:     $\sum_{n=1}^{\infty} \frac{2n+1}{3n+4} 5^n$

  • Step 1: General term:       $u_n = \frac{2n+1}{3n+4} 5^n$       $u_{n+1} = \frac{2(n+1)+1}{3(n+1)+4} 5^{n+1} = \frac{2n+3}{3n+7} 5^{n+1}$

  • Step 2: Compute the ratio:       $\frac{u_{n+1}}{u_n} = \frac{2n+3}{3n+7} 5^{n+1} \times \frac{3n+4}{(2n+1) 5^n} = \frac{2n+3}{2n+1} \times \frac{3n+4}{3n+7} \times 5$

  • Step 3: Calculate the limit:       $l = \lim_{n \to \infty} \frac{u_{n+1}}{u_n} = \lim_{n \to \infty} \left[\frac{2 + \frac{3}{n}}{2 + \frac{1}{n}} \times \frac{3 + \frac{4}{n}}{3 + \frac{7}{n}} \times 5\right] = 1 \times 1 \times 5 = 5$

  • Step 4: Evaluate: $l = 5 > 1$.

  • Step 5: By D'Alembert's Ratio Test, the series is divergent.

• Ratio Test Practice Problems:

  • 1. $\sum_{n=1}^{\infty} \frac{n^2}{3^n}$ : Convergent ($l = \frac{1}{3} < 1$)

  • 2. $\sum_{n=1}^{\infty} \frac{4 \cdot 7 \cdot 10 \cdots (3n+1)}{1 \cdot 2 \cdot 3 \cdots n}$ : Divergent ($l = 3 > 1$)

  • 3. $\sum_{n=1}^{\infty} \frac{3^n}{n!}$ : Convergent ($l = 0 < 1$)

  • 4. $\sum_{n=1}^{\infty} \frac{2n-1}{n^3 + a}$ : Convergent

  • 5. $\sum_{n=1}^{\infty} \frac{n!}{2^n + a}$ : Convergent (transcript lists as Convergent; mathematically diverges since $l = \infty$

  • 6. $\sum_{n=1}^{\infty} \frac{2^n}{n^3 + 1}$ : Divergent ($l = 2 > 1$)

  • 7. $\sum_{n=1}^{\infty} \frac{3n-1}{2^n}$ : Convergent ($l = \frac{1}{2} < 1$)

  • 8. $\sum_{n=1}^{\infty} \frac{3n-1}{n!}$ : Convergent ($l = 0 < 1$)

  • 9. $\sum_{n=1}^{\infty} \frac{n^n}{n!}$ : Divergent ($l = e > 1$)

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  10. $\sum_{n=1}^{\infty} \frac{2 \cdot 4 \cdot 6 \cdots (2n)}{5 \cdot 8 \cdot 11 \cdots (3n+2)}$ : Convergent ($l = \frac{2}{3} < 1$)

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  11. $\sum_{n=1}^{\infty} \frac{(n-1)!}{n^3}$ : Divergent ($l = \infty > 1$)

Systematic Diagnostic Algorithm for Series Convergence

• A structured decision process guides testing the convergence of positive-term series $\sum_{n=1}^{\infty} a_n$ with $a_n > 0$:

• Flowchart Rules and Pathways:

  • Initial Divergence Check: Check whether $a_n \to 0$ as $n \to \infty$.

  • If No: The series is Divergent by the $n^{\text{th}}$-term test.

  • If Yes or uncertain: Proceed to structural analysis of $a_n$.

  • Factorial or Exponential Check: Check if $a_n$ contains factorials ($n!$), exponentials ($c^n$), or power terms ($n^n$).

  • If Yes: Calculate $L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}$ via the Ratio Test.

    • If $L > 1$: Series is Divergent.

    • If $L < 1$: Series is Convergent.

    • If $L = 1$ or does not exist: Test Fails; select an alternate test.

  • If No: Check algebraic structure.

  • Polynomial Ratio Check: Check if $a_n$ is a ratio of polynomials.

  • If Yes: Apply the Limit Comparison Test with $v_n = \frac{1}{n^p}$.

  • If No: Apply the Direct Comparison Test.

Alternating Series and the Leibniz Test

• Definition: An alternating series is an infinite series whose successive terms alternate in sign, expressed as:   $\sum_{n=1}^{\infty} (-1)^{n-1} u_n = u_1 - u_2 + u_3 - u_4 + \dots$   where $u_n > 0$ for all $n$.

• Leibniz Test (Alternating Series Test):

  • An alternating series $\sum_{n=1}^{\infty} (-1)^{n-1} u_n$ is convergent if it satisfies two conditions:

  1. The terms are non-increasing in magnitude:        $u_1 > u_2 > u_3 > u_4 > \dots \quad \text{i.e., } u_{n+1} \le u_n \text{ for all } n$

  2. The limit of the general term is zero:        $\lim_{n \to \infty} u_n = 0$

Power Series and the Region of Convergence

• Definition of Power Series:

  • A power series in one variable centered at $c$ is an infinite series of the form:     $\sum_{n=0}^{\infty} a_n (x - c)^n = a_0 + a_1(x - c) + a_2(x - c)^2 + a_3(x - c)^3 + \dots$     where $a_n$ denotes the coefficient of the $n^{\text{th}}$ term and $c$ is a constant.

  • Properties:

  • The coefficients $a_0, a_1, a_2, \dots$ are independent of $x$.

  • When centered at the origin ($c = 0$), the power series simplifies to:       $\sum_{n=0}^{\infty} a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \dots$

• Fundamental Convergence Theorem:

  • For any power series $\sum_{n=0}^{\infty} a_n (x - a)^n$, exactly one of the following three mutually exclusive possibilities holds:

  1. The series converges only when $x = a$ (trivial convergence, radius $R = 0$).

  2. The series converges for all real values of $x$ (radius $R = \infty$; interval of convergence is $(-\infty, \infty)$).

  3. There exists a strictly positive real number $R$ such that the series converges absolutely for $|x - a| < R$ and diverges for $|x - a| > R$.

• Interval and Radius of Convergence:

  • The number $R$ is called the radius of convergence.

  • The interval of convergence comprises all values of $x$ for which the series converges.

  • Endpoint behavior: At the boundaries $x = a - R$ and $x = a + R$, the series may converge at both, diverge at both, or converge at one and diverge at the other; each boundary must be tested independently.

• Determining Convergence via the Ratio Test:

  • Consider $u_n = a_n (x - c)^n$ and $u_{n+1} = a_{n+1} (x - c)^{n+1}$.

  • Calculate the limit ratio:     $l = \lim_{n \to \infty} \left|\frac{u_{n+1}}{u_n}\right| = \lim_{n \to \infty} \left|\frac{a_{n+1} (x - c)^{n+1}}{a_n (x - c)^n}\right| = |x - c| \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$

  • Setting $M = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right|$:

  • The series converges when $|x - c| < M$.

  • The series diverges when $|x - c| > M$.

  • The radius of convergence is given by $R = M$, and the center of convergence is $c$.

• Power Series Worked Examples:

  • Example 1: Test convergence of:     $\frac{2}{1^3} x + \frac{3}{2^3} x^2 + \frac{4}{3^3} x^3 + \dots + \frac{n+1}{n^3} x^n + \dots \quad (x \in \mathbb{R})$

  • Terms: $u_n = \frac{n+1}{n^3} x^n$, $u_{n+1} = \frac{n+2}{(n+1)^3} x^{n+1}$.

  • Ratio limit:       $l = \lim_{n \to \infty} \left|\frac{u_{n+1}}{u_n}\right| = \lim_{n \to \infty} \left|\frac{n+2}{(n+1)^3} x^{n+1} \times \frac{n^3}{n+1} \frac{1}{x^n}\right|$       $l = |x| \lim_{n \to \infty} \left[\frac{n+2}{n+1} \times \frac{n^3}{(n+1)^3}\right] = |x| \times 1 \times 1 = |x|$

  • Conclusion: Convergent for $|x| < 1$; divergent for $|x| > 1$.

  • Example 2: Test convergence of the exponential series:     $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!} + \dots \quad (x \in \mathbb{R})$

  • Terms: $u_n = \frac{x^n}{n!}$, $u_{n+1} = \frac{x^{n+1}}{(n+1)!}$.

  • Ratio limit:       $l = \lim_{n \to \infty} \left|\frac{u_{n+1}}{u_n}\right| = \lim_{n \to \infty} \left|\frac{x^{n+1}}{(n+1)!} \times \frac{n!}{x^n}\right| = |x| \lim_{n \to \infty} \frac{1}{n+1} = |x| \times 0 = 0$

  • Conclusion: Since $l = 0 < 1$ regardless of $x$, the series is convergent for all values of $x \in \mathbb{R}$ ($R = \infty$).

  • Example 3: Test convergence of:     $\sum_{n=1}^{\infty} \frac{x^n}{x + n}$

  • Terms: $u_n = \frac{x^n}{x+n}$, $u_{n+1} = \frac{x^{n+1}}{x+n+1}$.

  • Ratio limit:       $l = \lim_{n \to \infty} \left|\frac{u_{n+1}}{u_n}\right| = \lim_{n \to \infty} \left|\frac{x^{n+1}}{x+n+1} \times \frac{x+n}{x^n}\right| = |x| \lim_{n \to \infty} \frac{n\left(1 + \frac{x}{n}\right)}{n\left(1 + \frac{x+1}{n}\right)} = |x| \times 1 = |x|$

  • Conclusion: Convergent for $|x| < 1$; divergent for $|x| > 1$.

• Power Series Practice Problems:

  • $\sum_{n=1}^{\infty} \frac{x^n}{n^4}$ : Convergent if $|x| < 1$, divergent if $|x| > 1$

  • $1 + 3x + 5x^2 + 7x^3 + \dots$ : Convergent if $|x| < 1$, divergent if $|x| > 1$

  • $1 + \frac{x}{2} + \frac{x^2}{2^2} + \frac{x^3}{2^3} + \dots$ : Convergent if $|x| < 2$, divergent if $|x| > 2$

  • $\sum_{n=1}^{\infty} \frac{x^n}{n}$ : Convergent if $|x| < 1$, divergent if $|x| > 1$

  • $\sum_{n=1}^{\infty} \frac{(n+1)x^n}{n^2}$ : Convergent if $|x| < 1$, divergent if $|x| > 1$

  • $\sum_{n=1}^{\infty} \frac{(x-3)^n}{n}$ : Convergent if $2 < x < 4$, divergent if $x > 4$ or $x < 2$

  • $\sum_{n=1}^{\infty} \frac{x^{2n}}{4^n}$ : Convergent if $|x| < 2$, divergent if $|x| > 2$