Exam 4 Notes

Section 10.1: Sequences

• A sequence is a list of numbers in a definite order: $a<em>1, a</em>2, a<em>3, a</em>4, …, a_n, …$

• Denoted as: $a<em>n$ or ${a</em>n}_{n=1}^{\infty}$

• Example: List the first three terms of the sequence starting at $n = 1$

  • a. $a_n = \frac{1-n}{n^2}$

  • b. a_n = (-1)^{n+1} \frac{2n-1}

• Example: List the first 5 terms of the recursive sequence defined below (starting at $n = 1$)

  • $a<em>1 = -2, a</em>{n+1} = \frac{na_n}{n+1}$

• Example: Find a formula for the general $a_n$ of the sequence, assuming that the pattern of the first few terms continues. (Assume you start with $n = 1$)

  • a. $0, 3, 8, 15, 24, …$

  • b. $1, -\frac{1}{4}, \frac{1}{9}, -\frac{1}{16}, \frac{1}{25}, …$

• Definition: The sequence $a<em>n$ converges to the number $L$ if for every positive number $\epsilon$ there corresponds an integer $N$ such that |an - L| < \epsilon whenever n > N.

  • If no such number $L$ exists we say that $a_n$ diverges.

  • If $a<em>n$ converges to $L$, we write $\lim</em>{n \to \infty} a<em>n = L$ or simply $a</em>n \to L$ and call $L$ the limit of the sequence.

• Example: Determine if the sequence converges or diverges and explain why.

  • a. $a_n = \frac{1-5n^4}{n^4+8n^3}$

  • b. $a_n = \frac{2n}{n+1}$

  • c. $a_n = \frac{2n+1}{1-3n}$

  • d. $a_n = \sin(\frac{\pi}{2} + \frac{1}{n})$

  • e. $a_n = \frac{\ln(n+1)}{\sqrt{n}}$

• If $P$ and $Q$ are highest powers of $n$ on top and bottom:

  • If P < Q, $\lim_{n \to \infty} = 0$

  • If P > Q, $\lim_{n \to \infty} = \infty$

  • If $P = Q$, $\lim_{n \to \infty} =$ ratio of coefficients

Section 10.2: Infinite Series

• Definition: Given a sequence of numbers $a<em>n$, an expression of the form $a</em>1 + a<em>2 + a</em>3 + \cdots + a_n + \cdots$ is an infinite series.

  • The number $a_n$ is the $n^{th}$ term of the series.

• The sequence $s_n$ defined by

  • $s<em>1 = a</em>1$

  • $s<em>2 = a</em>1 + a_2$

  • $\vdots$

  • $s<em>n = a</em>1 + a<em>2 + \cdots + a</em>n = \sum<em>{k=1}^{n} a</em>k$

• is the sequence of partial sums of the series, the number $s_n$ being the $n^{th}$ partial sum.

  • If the sequence of partial sums converges to a limit $L$, we say that the series converges and that its sum is $L$.

  • In this case, we write $a<em>1 + a</em>2 + \cdots + a<em>n + \cdots = \sum</em>{n=1}^{\infty} a_n = L$

  • If the sequence of partial sums of the series does not converge, we say that the series diverges.

• It is impossible to find a finite sum for something like: $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + \cdots + n + \cdots$

• But if we add the terms of the series $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \cdots + \frac{1}{2n} + \cdots$ we will get ever closer to a finite number. (This is called Zeno’s paradox)

  • $\sum_{n=1}^{\infty} \frac{1}{2n} = 1$

• An important example of an infinite series is the geometric series.

  • $a + ar + ar^2 + ar^3 + \cdots + ar^{n-1} + \cdots = \sum_{n=1}^{\infty} ar^{n-1}$

• Each term is obtained from the previous by multiplication of a common ratio that we will call $r$.

• Geometric Series:

  • If r < 1, the geometric series $a + ar + ar^2 + \cdots + ar^{n-1} + \cdots$ converges to $\frac{a}{1-r}$

  • \sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1 - r}, r < 1

  • If $r \geq 1$, the series diverges

• Ex: Write the following as a geometric series that starts at $n = 1$ and determine if it is convergent or divergent. If it converges, find its sum.

  • a. $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots$ (Zeno’s paradox)

  • b. $\frac{4}{9} - \frac{8}{27} + \frac{16}{81} - \frac{32}{243} + \frac{64}{729} - \cdots$

• Determine if the following geometric series are convergent or divergent. If they are convergent, find their sum.

  • a. $\sigma_{n=1}^{\infty} \frac{2^{n+1}}{5^n}$

  • b. $\sigma_{n=1}^{\infty} \frac{5^{2n}}{6^{n-1}}$

• Theorem 7: If $\sigma<em>{n=1}^{\infty} a</em>n$ converges, then $a_n \to 0$ as $n \to \infty$

  • Caution: Theorem 7 does not say that $\sigma<em>{n=1}^{\infty} a</em>n$ converges if $a<em>n \to 0$ as $n \to \infty$, it is possible for a series to diverge when $a</em>n \to 0$ as $n \to \infty$

• The $n^{th}$ term Test for Divergence (TFD)

  • $\sigma<em>{n=1}^{\infty} a</em>n$ diverges if $\lim<em>{n \to \infty} a</em>n$ fails to exist or is different from zero

• Ex: Determine whether the following series are convergent or divergent.

  • a. $\sigma_{n=1}^{\infty} \frac{n(n+1)}{n+2 \cdot n+3}$

  • b. $\sigma_{n=1}^{\infty} \frac{\cos(\pi n)}{5n}$

Section 10.3: The Integral Test

• In general, it is difficult to find the exact sum of a series.

  • Over the next few sections (10.3 – 10.6), we will use different tests to enable us to determine if a series is convergent or divergent without explicitly finding its sum.

• Theorem 9: The Integral Test

  • Let $a<em>n$ be a sequence of positive terms. Suppose that $a</em>n = f(n)$, where $f$ is a continuous, positive, decreasing function of $x$ for all $x \geq N$ ($N$ a positive integer).

  • Then the series $\sigma<em>{n=N}^{\infty} a</em>n$ and the integral $\int_{N}^{\infty} f(x) dx$ both converge or both diverge.

  • This idea comes from Section 8.8 in Calculus II.

  • The sum of the series is NOT equal to the value of the integral!

• A special case of The Integral Test is called p-series

  • The $p$-series $\sigma_{n=1}^{\infty} \frac{1}{n^p}$ converges if p > 1, diverges if $p \leq 1$

  • The harmonic series is a special case of the p-series where the p = 1.

  • $\sum_{n=1}^{\infty} \frac{1}{n}$

• Ex: Use p – series to determine whether the series is convergent or divergent.

  • a. $\sigma_{n=1}^{\infty} \frac{1}{n^2}$

  • b. $\sigma_{n=1}^{\infty} \frac{4}{n^{0.2}}$

• Ex: Use the Integral Test to determine whether the series is convergent or divergent.

  • $\sum_{n=1}^{\infty} \frac{n}{n^2 + 4}$

• Ex: Use the Integral Test to determine whether the series is convergent or divergent.

  • $\sum_{n=2}^{\infty} \frac{1}{n \ln(n)^2}$

• Ex: Use the Integral Test to determine whether the series is convergent or divergent.

  • $\sum_{n=1}^{\infty} \frac{e^n}{1 + e^{2n}}$

• Ex: Use the Integral Test to determine whether the series is convergent or divergent.

  • $\sum_{n=2}^{\infty} \frac{n + 4}{n^2 - 2n + 1}$

Section 10.4: Comparison Tests

• Theorem 10: Direct Comparison Test (DCT)

  • Let $\sigma a<em>n$ and $\sigma b</em>n$ be two series with $0 \leq a<em>n \leq b</em>n$ for all $n$.

    • If $\sigma b<em>n$ converges, then $\sigma a</em>n$ also converges

    • If $\sigma a<em>n$ diverges, then $\sigma b</em>n$ also diverges

• Ex: Use the Direct Comparison Test to determine whether the following series converge or diverge

  • $\sum_{n=1}^{\infty} \frac{1}{n^2 + 30}$

• Ex: Use the Direct Comparison Test to determine whether the following series converge or diverge

  • $\sum_{n=2}^{\infty} \frac{n + 2}{n^2 - n}$

• Ex: Use the Direct Comparison Test to determine whether the following series converge or diverge

  • $\sum_{n=1}^{\infty} \frac{\cos^2 n}{n^{\frac{3}{2}}}$

• Ex: Use the Direct Comparison Test to determine whether the following series converge or diverge

  • $\sum_{n=1}^{\infty} \frac{n}{3n + 1} \frac{1}{n}$

• Theorem 11 – Limit Comparison Test (LCT)

  • Suppose that an > 0 and bn > 0, for all $n \geq N$ ($N$ is an integer)

    • If $\lim<em>{n \to \infty} \frac{a</em>n}{b<em>n} = c$ and c > 0, then $\sigma a</em>n$ and $\sigma b_n$ both converge or both diverge

    • If $\lim<em>{n \to \infty} \frac{a</em>n}{b<em>n} = 0$ and $\sigma b</em>n$ converges, then $\sigma a_n$ converges

    • If $\lim<em>{n \to \infty} \frac{a</em>n}{b<em>n} = \infty$ and $\sigma b</em>n$ diverges, then $\sigma a_n$ diverges

• Ex: Use the Limit Comparison Test to determine whether the following series converge or diverge

  • $\sum_{n=1}^{\infty} \frac{n + 2}{n^3 - n^2 + 3}$

Examples of Limit Comparison Test

• Ex: Use the Limit Comparison Test to determine whether the following series converge or diverge

  • $\sum_{n=1}^{\infty} \frac{5n}{n \sqrt{4n}}$

• Ex: Use the Limit Comparison Test to determine whether the following series converge or diverge

  • $\sum_{n=2}^{\infty} \frac{1}{n \sqrt{n^4 - 1}}$

Section 10.5: Absolute Convergence; The Ratio and Root Tests

• Review on Factorials:

  • $3! = (3)(2)(1)$

  • $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$

  • $n! = n \cdot (n - 1) \cdot (n - 2) \cdot (n - 3) \cdots (2)(1)$

  • $(n + 1)! = (n + 1) \cdot n \cdot (n - 1) \cdots 2 \cdot 1$

  • $(2n + 3)! = (2n + 3) \cdot (2n + 2) \cdot (2n + 1) \cdot (2n) \cdots 2 \cdot 1$

  • $\frac{n!}{(n+1)!} = \frac{n \cdot (n-1) \cdot (n-2) \cdot(n-3) \cdots(2)(1)}{(n+1) \cdot n \cdot (n-1) \cdots 2 \cdot 1} = \frac{n!}{(n+1)n!} = \frac{1}{n+1}$

• Definition: A series $\sum a<em>n$ converges absolutely (is absolutely convergent) if the corresponding series of absolute values $\sum |a</em>n|$ converges

• Theorem 12 – The Absolute Convergence Test

  • If $\sum<em>{n=1}^{\infty} |a</em>n|$ converges, then $\sum<em>{n=1}^{\infty} a</em>n$ converges

• Theorem 13 – The Ratio Test

  • Let $\sum a_n$ be any series and suppose that

    • $\lim<em>{n \to \infty} \frac{|a</em>{n+1}|}{|a_n|} = \rho$

    • (a) the series converges absolutely if \rho < 1

    • (b) the series diverges if \rho > 1 or $\rho$ is infinite, and

    • (c) the test is inconclusive if $\rho = 1$

• Ex: Use the Ratio Test to determine whether the series converges or diverges.

  • $\sum_{n=1}^{\infty} \frac{2^n}{n!}$

• Ex: Use the Ratio Test to determine whether the series converges or diverges.

  • $\sum_{n=1}^{\infty} \frac{n!}{10^n}$

• Ex: Use the Ratio Test to determine whether the series converges or diverges.

  • $\sum_{n=1}^{\infty} \frac{e^{-n}}{n^3}$

Ratio Test Examples

• Ex: Use the Ratio Test to determine whether the series converges or diverges.

  • $\sum_{n=1}^{\infty} \frac{n!}{(2n + 1)!}$

• Theorem 14 – The Root Test

  • Let $\sum a<em>n$ be any series and suppose that $\lim</em>{n \to \infty} \sqrt[n]{|a_n|} = \rho$

    • (a) the series converges absolutely if \rho < 1

    • (b) the series diverges if \rho > 1 or $\rho$ is infinite, and

    • (c) the test is inconclusive if $\rho = 1$

Root Test Examples

• Ex: Use the Root Test to determine whether the series converges or diverges.

  • $\sum_{n=1}^{\infty} (\frac{4n + 3}{3n - 5})^n$

• Ex: Use the Root Test to determine whether the series converges or diverges.

  • $\sum_{n=1}^{\infty} (\frac{7}{2n + 5})^n$

Section 10.6: Alternating Series and Conditional Convergence

• A series whose terms alternate between positive and negative is called an alternating series

• Theorem 15 – The Alternating Series Test (AST)

  • The series $\sum<em>{n=1}^{\infty} (-1)^{n+1} u</em>n = u<em>1 - u</em>2 + u<em>3 - u</em>4 + \cdots$ Converges if the following conditions are satisfied:

    • The $u_n$’ s are all positive

    • The $u<em>n$’ s are eventually nonincreasing: $u</em>n \geq u_{n+1}$ for all $n \geq N$, for some integer $N$

    • $u_n \to 0$ as $n \to \infty$

Alternating Series Test Examples

• Ex: Use the Alternating Series Test to determine whether the series converges or diverges

  • $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}$

• Ex: Use the Alternating Series Test to determine whether the series converges or diverges

  • $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n^2 + 5}{n^2 + 4}$

• Test the series for convergence or divergence.

  • $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n^2 + 1}$

    • $u_n = \frac{n}{n^2 + 1}$

    • $u_n$ non-increasing

    • $\lim<em>{n \to \infty} u</em>n = 0$

    • Converges by AST

• Definition: A series that is convergent, but not absolutely convergent is called conditionally convergent.

• Ex: Determine if the series converges absolutely, converges conditionally, or diverges.

  • $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n^3 + 1}$

Determine Absolute vs Conditional Convergence Examples

• Ex: Determine if the series converges absolutely, converges conditionally, or diverges.

  • $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n + 3}$

• Ex: Determine if the series converges absolutely, converges conditionally, or diverges.

  • $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n!}{2^n}$

    • Using the Ratio Test, it diverges.

Summary of Tests to Determine Convergence or Divergence

• The $n^{th}$ – Term Test for Divergence: Unless $a_n \to 0$ as $n \to \infty$, the series diverges

• Geometric Series: $\sigma ar^{n-1}$ converges if r < 1; otherwise, it diverges

• $p$-series: $\sigma \frac{1}{n^p}$ converges if p > 1; otherwise, it diverges

• Series with nonnegative terms: Try the Integral, Direct Comparison, Limit Comparison, Ratio, or Root Tests.

• Series with some negative terms: Try the Ratio or Root Tests. Remember that absolute convergence implies convergence.

• Alternating Series: $\sigma a_n$ converges if the series satisfies the conditions of the Alternating Series Test

Section 10.7: Power Series

• Definition: A power series about $x = 0$ is a series of the form $\sum<em>{n=0}^{\infty} c</em>nx^n = c<em>0 + c</em>1x + c<em>2x^2 + \cdots + c</em>nx^n + \cdots$

• A power series about $x = a$ is a series of the form$\sum<em>{n=0}^{\infty} c</em>n(x - a)^n = c<em>0 + c</em>1(x - a) + c<em>2(x - a)^2 + \cdots + c</em>n(x - a)^n + \cdots$

  • in which the center $a$ and the coefficients $c<em>0, c</em>1, c<em>2, …, c</em>n, …$ are constants. Functions can be written as power series shown above.

  • So the big question becomes, “Where do these functions (power series) converge?”

• The number $R$ in case (1) is called the radius of convergence of the power series. (This is where the Ratio Test is needed and possibly the Root Test)

• The interval of convergence of a power series is the interval that consists of all values of $x$ for which the series converges.

• Corollary: The convergence of the series $\sigma c_n(x - a)^n$ is described by one of the following three cases:

  • There is a positive number $R$ such that the series diverges for $x$ with |x - a| > R but converges absolutely for $x$ with |x - a| < R. The series may or may not converge at either of the endpoints $x = a - R$ and $x = a + R$.

  • The series converges absolutely for every $x$ $(R = \infty)$.

  • The series converges at $x = a$ and diverges elsewhere $(R = 0)$.

Intervals of Convergence Possibilities

• Converges on $[a-R, a +R]$

• Converges on $(a− R, a +R]$

• Converges on $[a-R, a + R)$

• Converges on $(a− R, a + R)$

• Converges everywhere

• Converges only at x = a

How to Test a Power Series for Convergence

• Use the Ratio Test or the Root Test to find the largest open interval where the series converges absolutely,

  • |x − a| < R or a − R < x < a + R.

  • If $R$ is finite, test for convergence or divergence at each endpoint.

  • If $R$ is finite, the series diverges for |x − a| > R.

Power and Intervals of Convergence Examples

• Ex: Find the radius of convergence and interval of convergence of the series.

  • $\sum_{n=0}^{\infty} \frac{nx^n}{n + 2}$

Power Series Examples

• Ex: Find the radius of convergence and interval of convergence of the series.

  • $\sum_{n=1}^{\infty} \frac{x^n}{n^2 + 3}$

• Ex: Find the radius of convergence and interval of convergence of the series.

  • $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x + 2)^n}{n \cdot 2^n}$

• Ex: Find the radius of convergence and interval of convergence of the series.

  • $\sum_{n=0}^{\infty} \frac{n!}{x^n}$

• Ex: Find the radius of convergence and interval of convergence of the series.

  • $\sum_{n=1}^{\infty} nn!x^n$

Section 10.8: Taylor and Maclaurin Series

• Definition: Let $f$ be a function with derivatives of all orders throughout some interval containing $a$ as an interior point. Then the Taylor series generated by $f$ at $x = a$ is

  • $\sum_{k=0}^{\infty} \frac{f^k(a)}{k!} (x - a)^k = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!} (x - a)^2 + \cdots + \frac{f^n(a)}{n!} (x - a)^n + \cdots$

• The Maclaurin series of $f$ is the Taylor series generated by $f$ at $x = 0$, or

  • $\sum_{k=0}^{\infty} \frac{f^k(0)}{k!} x^k = f(0) + f'(0)x + \frac{f'(0)}{2!} x^2 + \cdots + \frac{f^n(0)}{n!} x^n + \cdots$

• ***$f^k$ is the kth derivative of $f$ NOT $f$ raised to the k power ***

• Definition: Let $f$ be a function with derivatives of order $k$ for $k = 1, 2, …, N$ in some interval containing $a$ as an interior point. Then, for any integer $n$ from 0 through $N$, the Taylor polynomial of order $n$ generated by $f$ at $x = a$ is the polynomial

  • $P_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!} (x - a)^2 + \cdots + \frac{f^k(a)}{k!} (x - a)^k + \cdots + \frac{f^n(a)}{n!} (x - a)^n$

• On the next slide, we will see a couple of examples of how accurate a Taylor polynomial can be to a function

Taylor Polynomials Visualization

• Graphs of $y = e^x$ along with various Taylor Polynomials is shown

• Graphs of $y = \cos(x)$ along with various Taylor Polynomials is shown

Taylor Polynomial Examples

• Ex: Find the Taylor Polynomial $P_3(x)$ for the function $f$ centered at the number $a$.

  • $f(x) = e^{2x}, a = 0$

• Ex: Find the Taylor Polynomial $P_3(x)$ for the function $f$ centered at the number $a$.

  • $f(x) = \sin(x), a = \frac{\pi}{3}$

• Ex: Find the Taylor Polynomial $P_3(x)$ for the function $f$ centered at the number $a$.

  • $f(x) = \sqrt{x}, a = 4$

• Ex: Find the Taylor Polynomial $P_3(x)$ for the function $f$ centered at the number $a$.

  • $$f(x) = 2^x, a = 1