Deck 10: Series

Question 1

Use the nth term test to investigate the series $\sum_{n=1}^{\infty} \frac{n+1}{n+2}$ .

A) The series converges.
B) $\lim _{\mathrm{n} \rightarrow} \frac{\mathrm{n}+1}{\mathrm{n}+2}=1 \neq 0$ , so the series diverges.
C) $\lim _{\mathrm{n} \rightarrow} \frac{\mathrm{n}+1}{\mathrm{n}+2}=0$ , so the test fails to tell us anything about the series.
D) $\lim _{\mathrm{n} \rightarrow} \frac{\mathrm{n}+1}{\mathrm{n}+2}=1$ , so the test fails to tell us anything about the series.

Answer 1

$\lim _{\mathrm{n} \rightarrow} \frac{\mathrm{n}+1}{\mathrm{n}+2}=1 \neq 0$ , so the series diverges.

Question 2

Use the integral test to investigate the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$ .

A) The integral $\int_{1}^{\infty} \frac{1}{x^{3 / 2}} d x=2$ , so the series $\sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}}=2$ .
B) The integral $\int_{1}^{\infty} \frac{1}{\mathrm{x}^{3 / 2}} \mathrm{dx}=2$ , so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$ converges.
C) The integral $\int_{1}^{\infty} \frac{1}{\mathrm{x}^{3 / 2}} \mathrm{dx}=2$ , so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$ diverges.
D) The integral $\int_{1}^{\infty} \frac{1}{x^{3 / 2}} d x=2$ , so the test fails to tell us anything about the series.

Answer 2

The integral $\int_{1}^{\infty} \frac{1}{\mathrm{x}^{3 / 2}} \mathrm{dx}=2$ , so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$ converges.

Question 3

Use the ratio test to investigate the series $\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$ .

A) $\lim _{n \rightarrow \infty} \frac{\frac{2^{n+1}}{(n+1) !}}{\frac{2 n}{n !}}=\lim _{n \rightarrow \infty} \frac{2}{n+1}=0$ , so the series $\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$ diverges.
B) $\lim _{n \rightarrow \infty} \frac{\frac{2^{n+1}}{(n+1) !}}{\frac{2 n}{n !}}=\lim _{n \rightarrow} \frac{2}{n+1}=0<1$ , so the series $\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$ converges.
C) $\lim _{n \rightarrow \infty} \frac{\frac{2^{n+1}}{(n+1) !}}{\frac{2 n}{n !}}=\lim _{n \rightarrow \infty} \frac{2}{n+1}=0$ , so the ratio test fails to tell us anything about the series.
D) $\lim _{n \rightarrow \infty} \frac{\frac{2 n}{n !}}{\frac{2^{n+1}}{(n+1) !}}=\lim _{n \rightarrow \infty} \frac{n+1}{2}=\infty$ , so the series $\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$ converges.

Answer 3

$\lim _{n \rightarrow \infty} \frac{\frac{2^{n+1}}{(n+1) !}}{\frac{2 n}{n !}}=\lim _{n \rightarrow} \frac{2}{n+1}=0<1$ , so the series $\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$ converges.

Question 4

Investigate the alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ .

A) The p- series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ converges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ converges absolutely.
B) The p-series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ diverges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ converges conditionally.
C) The ratio test gives $\mathrm{r}=1$ , so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{(-1)^{\mathrm{n}+1}}{\mathrm{n}^{2}}$ diverges.
D) The p-series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ diverges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ diverges.

Question 5

Find the interval of convergence of the power series $\sum_{n=1}^{\infty} \frac{x^{n}}{n}$ .

A)
B) $-1 \leq x<1$
C) $-1 \leq x \leq 1$
D)

Question 6

Find a Maclaurin series expansion forf(x) $=e^{3 x}$ .

A) $\sum_{n=0}^{\infty} \frac{x^{n}}{n !}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots$
B) $\sum_{n=0}^{\infty} \frac{(3 \mathrm{x})^{n}}{n !}=1+3 \mathrm{x}+\frac{9 \mathrm{x}^{2}}{2 !}+\frac{27 \mathrm{x}^{3}}{3 !}+\ldots$
C) $\sum_{n=0}^{\infty} \frac{(3 x)^{n}}{(3 n) !}=1+\frac{3 x}{3 !}+\frac{9 x^{2}}{6 !}+\frac{27 x^{3}}{9 !}+\ldots$
D) $\sum_{n=0}^{\infty} \frac{3 x^{n}}{n !}=3+3 x+\frac{3 x^{2}}{2 !}+\frac{3 x^{3}}{3 !}+\ldots$

Question 7

Find a Taylor series expansion for $\mathrm{f}(\mathrm{x})=\sin \mathrm{x}$ with $\mathrm{a}=\pi$ .

A) $\sum_{n=0}^{\infty} \frac{(x-\pi)^{2 n+1}}{(2 n+1) !}=(x-\pi)+\frac{(x-\pi)^{3}}{3 !}+\frac{(x-\pi)^{5}}{5 !}+\frac{(x-\pi)^{7}}{7 !}+\ldots$
B) $\sum_{n=0}^{\infty} \frac{(-1)^{n}(\mathrm{x}-\pi)^{2 \mathrm{n}+1}}{(2 \mathrm{n}) !}=1-\frac{(\mathrm{x}-\pi)^{2}}{2 !}+\frac{(\mathrm{x}-\pi)^{4}}{4 !}-\frac{(\mathrm{x}-\pi)^{6}}{6 !}+\ldots$
C) $\sum_{n=0}^{\infty} \frac{(-1)^{n}(\mathrm{x}-\pi)^{2 \mathrm{n}+1}}{(2 \mathrm{n}+1) !}=(\mathrm{x}-\pi)-\frac{(\mathrm{x}-\pi)^{3}}{3 !}+\frac{(\mathrm{x}-\pi)^{5}}{5 !}-\frac{(\mathrm{x}-\pi)^{7}}{7 !}+\ldots$
D) $\sum_{n=0}^{\infty} \frac{(-1)^{n+1}(x-\pi)^{2 n+1}}{(2 n+1) !}=-(x-\pi)+\frac{(x-\pi)^{3}}{3 !}-\frac{(x-\pi)^{5}}{5 !}+\frac{(x-\pi)^{7}}{7 !}-\ldots$

Question 8

Use the first four non- zero terms of the Maclaurin series for $f(x)=e^{x}$ to estimate $e^{-0.3}$ .

A) .7405
B).7399
C) .7407
D).7402

Question 9

Find the Fourier series for the square wave (of period $2 \pi$ ) given by $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}-1, & -\pi \leq \mathrm{x}<0 \\ 1, & 0 \leq \mathrm{x} \leq \pi\end{array}\right.$

A) $\sum_{n=0}^{\infty} \frac{2}{\pi} \cdot \frac{\sin [(2 n+1) x]}{2 n+1}=\frac{2}{\pi}\left[\sin x+\frac{\sin 3 x}{3}+\frac{\sin 5 x}{5}+\frac{\sin 7 x}{7}+\ldots\right]$
B) $\sum_{n=0}^{\infty} \frac{4}{\pi} \cdot \frac{\sin [(2 n+1) x]}{2 n+1}=\frac{4}{\pi}\left[\sin x+\frac{\sin 3 x}{3}+\frac{\sin 5 x}{5}+\frac{\sin 7 x}{7}+\ldots\right]$
C) $\sum_{n=0}^{\infty} \frac{4}{\pi} \cdot \frac{\cos [(2 n+1) x]}{2 n+1}=\frac{4}{\pi}\left[\cos x+\frac{\cos 3 x}{3}+\frac{\cos 5 x}{5}+\frac{\cos 7 x}{7}+\ldots\right]$
D) $\sum_{n=0}^{\infty} \frac{2}{\pi} \cdot \frac{\cos [(2 n+1) x]}{2 n+1}=\frac{2}{\pi}\left[\cos x+\frac{\cos 3 x}{3}+\frac{\cos 5 x}{5}+\frac{\cos 7 x}{7}+\ldots\right]$

Question 10

For the problems below, determine whether each series converges or diverges.

- $1+\frac{1}{8}+\frac{1}{27}+\ldots+\frac{1}{n^{3}}+\ldots$

Question 11

For the problems below, determine whether each series converges or diverges.

- $\sum_{n=1}^{\infty} \frac{1}{(3 n-1)^{2}}$

Question 12

For the problems below, use either the ratio test or the integral test to determine whether each series converges or diverges.

- $\sum_{n=1}^{\infty} \frac{\mathrm{n}+4}{\mathrm{n} \cdot 5^{\mathrm{n}}}$

Question 13

For the problems below, use either the ratio test or the integral test to determine whether each series converges or diverges.

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

Question 14

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally.

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

Question 15

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally.

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

Question 16

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally.

- $\sum_{\mathrm{n}=1}^{\infty}(-1)^{\mathrm{n}+1} \frac{1}{\sqrt[3]{\mathrm{n}}}$

Question 17

For the problems below, find the interval of convergences of each series.

- $\sum_{n=1}^{\infty} \frac{(6 x)^{n}}{(3 n) !}$

Question 18

For the problems below, find the interval of convergences of each series.

-18 $\sum_{\mathrm{n}=1}^{\infty} \frac{(-1)^{\mathrm{n}}(\mathrm{x}-5)^{\mathrm{n}}}{\sqrt[3]{\mathrm{n}}}$

Question 19

For the problems below, find a Maclaurin series expansion for each function. Include at least thereetarms.

- $f(x)=e^{2 x}$

Question 20

For the problems below, find a Maclaurin series expansion for each function. Include at least thereetarms.

- $f(x)=\sin 3 x$

Question 21

Find a Maclaurin series expansion for $f(x)=\frac{e^{x}-1}{x}$ .

Question 22

Evaluate $\int_{0}^{1} \cos \sqrt{\mathrm{x}} \mathrm{dx}$ . (Use three non-zero terms.) Round to four significant digits.

Question 23

For the problems below, find the Taylor series expansion for each function for the given value of a. Give at least three terms.

- $f(x)=\frac{2}{x^{3}}, a=2$

Question 24

For the problems below, find the Taylor series expansion for each function for the given value of a. Give at least three terms.

- $f(x)=\cos x, a=\frac{\pi}{4}$

Question 25

Calculate the value of $\mathrm{e}^{1.2}$ using the first four non- zero terms of a Taylor series. Round to five significant digits.

Question 26

Find the Fourier series expansion of $f(x)=3 x, 0 \leq x<2 \pi$ . Write at least four terms.