University Study Notes: Functions, Sequences, and Series

Fundamental Tasks and Problems Involving Functions

• Evaluation: Given a value $x_0$, determine the value of the function at that point, $f(x_0)$. This involves challenges when dealing with irrational exponents or roots, such as evaluating $f(x) = x^{1/2}$, $f(x) = x^{2/3}$, or $f(x) = 10^x$ for all real $x$.
  • Examples of complex evaluation include finding the exact value of $\sqrt{\sqrt{2}}$ or $10^{\sqrt{2}}$.
• Plotting and Continuity: This involves graphing $f(x)$ versus $x$. Key considerations include:
  • How to join discrete points to form a continuous curve.
  • Maintaining a "smooth" curve and the physical justification for doing so.
  • Situations where smoothness is expected include physical quantities like heat, temperature, pressure, current, voltage, and magnetic or electric fields.
  • Non-smooth examples, such as the absolute value function $y = |x|$, which has a sharp corner at the origin.
• Inversion: Given a value $y_0$, find the corresponding $x$ such that $y = f(x_0)$.
• Growth, Fall, and Critical Points:
  • Increasing Behavior: The function increases in the intervals $x < x_1$, $x_2 < x < x_3$, and $x > x_4$.
  • Decreasing Behavior: The function decreases in the intervals $x_1 < x < x_2$ and $x_3 < x < x_4$.
  • Critical Points: These are points $x ∈ \mathbb{R}$ where the graph of $f(x)$ attains a maximum or minimum value. At these points, the function hits a peak or trough and changes its growth/fall behavior.
  • Asymptotics: Observing the behavior as $x$ reaches extreme values. For large positive $x$, the function may tend toward a non-zero value; for large negative $x$, it may tend toward zero.

Sequences: Definitions and Examples

• Formal Definition: A sequence of real numbers is a function defined from the set of non-negative integers $\{0, 1, 2, ...\}$ to the set of real numbers $\mathbb{R}$.
  • Domain/Codomain Notation: $f: \mathbb{N} \cup \{0\} \rightarrow \mathbb{R}$ or $f: \mathbb{N} \rightarrow \mathbb{R}$.
• Basic Sequence Examples:
  • $a_n = n$: Generates the sequence $0, 1, 2, 3, ...$
  • $a_n = (-1)^n$: Generates the sequence $1, -1, 1, -1, ...$
  • $a_n = \frac{1}{n+1}$: Generates the sequence $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ...$
• Recursive Sequences:
  • Fibonacci Sequence: Defined by $a_0 = 0$, $a_1 = 1$, and $a_{n+1} = a_n + a_{n-1}$ for $n \ge 1$. The sequence progresses as $0, 1, 1, 2, 3, 5, 8, 13, 21, ...$
  • Arithmetic Progression (AP): Defined by $a_{n+1} = a_n + d$ for $n \ge 0$, where $d$ is the common difference. The $n$-th term is $a_n = a_0 + nd$.
  • Geometric Progression (GP): Defined by $a_{n+1} = a_n r$ for $n \ge 0$, where $r$ is the common ratio. The $n$-th term is $a_n = a_0 r^n$.
• The $\sqrt{2}$ Sequence (Newton-Raphson approach):
  • Formula: $a_0 = 2$, $a_{n+1} = \frac{a_n}{2} + \frac{1}{a_n}$ for $n \ge 0$.
  • Calculated terms:
    • $a_0 = 2$
    • $a_1 = 1.5$
    • $a_2 = 1.416666...$
    • $a_3 = \frac{577}{408} = 1.414215686...$
  • Target value ($\sqrt{2}$): $1.4142135623730951...$
  • Significance: While $\sqrt{2}$ is irrational and cannot be expressed as a fraction $\frac{p}{q}$, individual terms in this sequence ($a_n$) are rational and can be computed precisely to approximate $\sqrt{2}$ as $n$ increases.
• Visualizing Sequences: Sequences are often visualized using stem plots to show discrete values for each integer $n$. Different sequence types exhibit distinct patterns: increasing, alternating, and decreasing.

Series: Sequences Defined as Summations

• Definition: Given a sequence $\{a_n\}$, a summation sequence $\{s_n\}$ is defined where each term represents the "partial sum" of the sequence up to that point.
  • $s_1 = a_1$
  • $s_{n+1} = s_n + a_{n+1}$
  • General form: $s_n = a_1 + a_2 + ... + a_n = \sum_{k=1}^{n} a_k$
• Key Series Examples:
  • Sum of Natural Numbers: For $a_n = n$, the partial sum is $s_n = \frac{n(n+1)}{2}$. The sequence of partial sums is $1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...$
  • Harmonic Series ($H_n$): $H_n = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} = \sum_{k=1}^{n} \frac{1}{k}$.
  • Generalized Harmonic Series ($H_n^{(2)}$): $H_n^{(2)} = 1 + \frac{1}{2^2} + \frac{1}{3^2} + ... + \frac{1}{n^2} = \sum_{k=1}^{n} \frac{1}{k^2}$.
  • Alternating Harmonic Series: $s_n = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... + \frac{(-1)^{n-1}}{n} = \sum_{k=1}^{n} \frac{(-1)^{k-1}}{k}$.
  • Madhava-Leibniz $\pi/4$ Series: $s_n = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + ... + \frac{(-1)^{n-1}}{2n-1} = \sum_{k=1}^{n} \frac{(-1)^{k-1}}{2k-1}$.

Boundedness of Sequences

• Bounded Above: A sequence $\{a_n\}$ is bounded above if there exists a constant $M$ such that $a_n \le M$ for all $n$.
• Bounded Below: A sequence $\{a_n\}$ is bounded below if there exists a constant $m$ such that $a_n \ge m$ for all $n$.
• Analysis of Examples:
  • $a_n = n$: Lower bound is $0$, no upper bound (unbounded above).
  • $a_n = (-1)^n$: Lower bound is $-1$, upper bound is $1$.
  • $a_n = \frac{1}{n+1}$: Lower bound is $0$, upper bound is $1$.
  • $a_n = -2^n$: No lower bound, upper bound is $-1$.
  • $a_n = (-2)^n$: No lower bound, no upper bound (unbounded).
  • Arithmetic Progression ($a_n = a_0 + nd$):
    • If $d > 0$: Lower bound is $a_0$, unbounded above.
    • If $d < 0$: Unbounded below, upper bound is $a_0$.
  • Geometric Progression ($a_n = 2r^n$):
    • If $r > 1$: Lower bound is $2$, unbounded above.
    • If $0 < r < 1$: Lower bound is $0$, upper bound is $2$.
  • $\sqrt{2}$ sequence ($a_0 = 2$): Lower bound is $1$, upper bound is $2$.

Monotonicity of Sequences and Series

• Increasing: $a_{n+1} > a_n$ for all $n$.
• Non-decreasing: $a_{n+1} \ge a_n$ for all $n$.
• Decreasing: $a_{n+1} < a_n$ for all $n$.
• Non-increasing: $a_{n+1} \le a_n$ for all $n$.
• Monotonicity Specifics:
  • Arithmetic Progression ($d > 0$): Increasing.
  • Arithmetic Progression ($d < 0$): Decreasing.
  • Geometric Progression ($r > 1$): Increasing.
  • Geometric Progression ($0 < r < 1$): Decreasing.
  • $\sqrt{2}$ sequence: Decreasing (flattening toward $\sqrt{2}$).
  • $s_n = 1 + 2 + ... + n$: Increasing.
  • Harmonic Series ($H_n$) and variants ($H_n^{(2)}$): Increasing.
  • Alternating Series (Harmonic and Madhava-Leibniz): Neither strictly increasing nor strictly decreasing (non-monotonic).

Concepts of Limits

• Formal Definition (Epsilon-N): A sequence $\{a_n\}$ tends to a limit $L$ if for any ̵ > 0, there exists an integer $N$ such that for all $n > N$, the term lies within the range L - ̵ \le a_n \le L + ̵.
  • Notation: $\lim_{n \to \infty} a_n = L$
• Convergence and Divergence Definitions:
  • Convergent: The sequence has a finite limit.
  • Diverges to $\infty$: The sequence grows unbounded and remains positive.
  • Diverges to $-\infty$: The sequence grows unbounded in the negative direction.
  • No Limit: The sequence does not stabilize at a finite value and does not grow unbounded in a single direction (e.g., oscillating sequences).
• Examples of Limits ($n \to \infty$):
  • $a_n = 1 \rightarrow$ Limit is $1$.
  • $a_n = \frac{1}{n} \rightarrow$ Limit is $0$.
  • $a_n = n \rightarrow$ Diverges to $\infty$.
  • $a_n = -2^n \rightarrow$ Diverges to $-\infty$.
  • $a_n = (-1)^n \rightarrow$ No limit (oscillates between $-1$ and $1$).
  • $a_n = (-2)^n \rightarrow$ No limit (oscillates with increasing magnitude).

Questions & Discussion

• Question (Sequence Identification): Which of the following functions represents a sequence?

  • $f_1: \mathbb{R} \rightarrow \mathbb{R}$
  • $f_2: \mathbb{N}_0 \rightarrow \mathbb{R}$
  • $f_3: \mathbb{R}^- \rightarrow \mathbb{R}$
  • $f_4: \mathbb{R}^+ \rightarrow \mathbb{R}$
  • $f_5: \mathbb{N}_0 \rightarrow \mathbb{Z}$
  • Answer: By definition, functions whose domain is the set of natural numbers (including 0), such as $f_2$ and $f_5$, qualify as sequences.

• Question (Arithmetic Progression): What is the common difference in the arithmetic progression $5, 14, 23, 32, 41, 50, ...$?

  • Answer Calculation: $14 - 5 = 9$. The common difference $d = 9$.

• Question (Recursion Identification): Consider the sequence $2, 2 - \frac{1}{2}, 2 - \frac{1}{2 - \frac{1}{2}}, 2 - \frac{1}{2 - \frac{1}{2 - \frac{1}{2}}}, ...$. What is the recursion formula?

  • Analysis:
    • $a_0 = 2$
    • $a_1 = 2 - \frac{1}{a_0} = 2 - \frac{1}{2} = 1.5$
    • $a_2 = 2 - \frac{1}{a_1}$
    • The correct recursion is $a_{n+1} = 2 - \frac{1}{a_n}$.

• Question (Partial Sum Analysis): Consider a series $S = \sum_{n=1}^{\infty} \frac{1}{(2n+1)3^n}$. Which option describes the partial sum $S_n$?

  • Answer: $S_n = \frac{1}{3 \cdot 3} + \frac{1}{5 \cdot 3^2} + \frac{1}{7 \cdot 3^3} + ... + \frac{1}{(2n+1)3^n}$. Specifically, for the first term when $n=1$, we have $\frac{1}{(2(1)+1)3^1} = \frac{1}{3 \cdot 3} = \frac{1}{9}$. The correct representation corresponds to Option D in the provided choices: $S_n = \frac{1}{3 \cdot 3} + \frac{1}{5 \cdot 3^2} + \frac{1}{7 \cdot 3^3} + ... + \frac{1}{(2n+1)3^n}$.

• Question (Monotonicity Check): Which of the following is correct?

  • Statement: The sequence $a_n = -\frac{1}{n}$ is an increasing sequence.
  • Verification: As $n\rightarrow\infty$, $\frac{1}{n}$ decreases, making its negative counterpart $-\frac{1}{n}$ increase towards zero. Thus, the statement is correct.