Deck 37: Infinite Series

Question 1

Use the ratio test to determine if the following series converges or diverges: $4+8+\frac{64}{3}+\ldots+\frac{4^{n}}{n}+\ldots$

Answer 1

converges

Question 2

Use the partial sum test to determine if the series converges: the geometric series: $72+36+18+9$ $+\ldots$

Answer 2

converges

Question 3

Use the ratio test to determine if the series converges: $1+\frac{1}{2}+\frac{2 !}{4}+\frac{3 !}{8}+\ldots+\frac{n !}{2^{n}}+\ldots$

Answer 3

diverges

Question 4

Use the ratio test to determine if the series converges: $1+\frac{3}{2}+\frac{9}{8}+\frac{27}{48}+\ldots+\frac{3^{n}}{2^{n} n !}+\ldots$

Question 5

Use the ratio test to determine if the following series converges or diverges: $\frac{6}{5}+\frac{12}{25}+\frac{24}{125}+\frac{48}{625}+\ldots$

Question 6

Use the partial sum test to determine if the series converges of diverges: $9+3+1+\ldots$

Question 7

Use the limit test to determine if the series converges of diverges: $1+\frac{2}{3}+\frac{4}{9}+\ldots$

Question 8

Use the limit test to determine if the following series converges or diverges: $1+1.5+2.25+3.375+\ldots$

Question 9

Use the partial sum test to determine if the series converges or diverges: $5+4+3.2+\ldots$

Question 10

Use the ratio test to determine if the series converges of diverges:
$2+2+\frac{4}{3}+\frac{2}{3}+\ldots+\frac{2^{n}}{n !}+\ldots$

Question 11

Compute the following number, to three decimal places, using three terms of the appropriate Maclaurin's series: $\sqrt[3]{-1.5}$

Question 12

Use the ratio test to find the interval of convergence of the following power series:
$x^{2}-\frac{x^{4}}{4}+\frac{x^{6}}{9}+\ldots+\frac{(-1)^{n+1} x^{2 n}}{n^{2}}+\ldots$

Question 13

Use the ratio test to find the interval of convergence of the power series: $\frac{x}{9}+\frac{2 x^{2}}{27}+\frac{3 x^{3}}{81}+\ldots+\frac{n x^{n}}{3^{n+1}}$

Question 14

Compute the number to three decimal places using three terms of the appropriate Maclaurin series:
$\sqrt[3]{e^{2}}$

Question 15

Find the first four terms of the Maclaurin series for the function: $e^{2 x} \cos x$

Question 16

Find the first four terms of the Maclaurin series for the function: $f(x)=\ln (3 x+1)$

Question 17

Find the first four terms of the Maclaurin series for the function: $y=x^{2} e^{x}$

Question 18

Use the ratio test to find the interval of convergence of the power series:
$x+\frac{x^{2}}{2}+\frac{x^{3}}{4}+\ldots+\frac{x^{n}}{2^{n-1}}+\ldots$

Question 19

Compute the following number, to three decimal places, using three terms of the appropriate Maclaurin's series: $\ln (0.8)$

Question 20

Compute the value of the following expression, to three decimal places, using three terms of the appropriate Taylor's series: $\cos 40^{\circ}$

Question 21

Compute the value of the following expression, to three decimal places, using three terms of the appropriate Taylor's series: $\sqrt{1.25}$

Question 22

Compute the value of $e^{1.2}$ , to three decimal places, using three terms of the Taylor's series expanded about $\mathrm{a}=1$ .

Question 23

Compute the value of the following expression, to three decimal places, using three terms of the appropriate Taylor's series: $\frac{1}{1.05}$

Question 24

Compute the value of the following expression, to three decimal places, using three terms of the appropriate Taylor's series: $\ln 0.97$

Question 25

Compute the value of the following expression, to three decimal places, using three terms of the appropriate Taylor's series: $\sin 0.4$

Question 26

Compute the value of $\cos \frac{\pi}{3}$ , to three decimal places, using three terms of the Taylor's series expanded about $a=\frac{\pi}{4}$ .

Question 27

Use the series for $\cos x$ to find the series for $\cos x^{2}$ .

Question 28

Use the series for $\cos x$ to find the series for $\cos 2 x$ .

Question 29

Add the appropriate series to obtain the series for $\sin x+\cos^{2} x$ .

Question 30

Multiply the appropriate series to obtain the series for $e^{2 x} \cos 2 x$ .

Question 31

Find the series for $x e^{x^{2}}$ by differentiating $e^{x^{2}}$ and multiplying by the appropriate factor.

Question 32

Use the series for $\cos x$ to find the series for $\cos \sqrt{x}$ .

Question 33

Find the series for $\frac{x}{(1+x)^{2}}$ by differentiating $\frac{1}{1+x}$ and multiplying the related series by an appropriate factor.

Question 34

Add the appropriate series to obtain the series for $e^{2 x}+\sin^{2} x$ .

Question 35

Multiply the appropriate series to obtain the series for $e^{x} \sqrt{1+x}$ .

Question 36

Evaluate the integral to three decimal places by integrating the first three terms of the series: $\int_0^2cos x dx$

Question 37

Evaluate the integral to three decimal places by integrating the first three terms of the series: $\int_{0}^{1 / 4} e^{\sqrt{x}} d x$

Question 38

Add the appropriate series to obtain the series for $\ln (1+x)+\sin x$ .

Question 39

Multiply the appropriate series to obtain the series for $\frac{\sqrt{1+x}}{1+x}$ .

Question 40

Evaluate the integral to three decimal places by integrating the first three terms of the series:
$\int_{1}^{2} \frac{\cos x}{x} d x$

Question 41

Write a Fourier series for the function below:

Question 42

Write seven terms of the Fourier series given the following coefficients: $a_{0}=8, a_{1}=4, a_{2}=3, a_{3}=1$ , $b_{1}=5, b_{2}=4, b_{3}=2$

Question 43

Write seven terms of the Fourier series given the following coefficients: $a_{0}=2.4, a_{1}=7.8, a_{2}=4.65$ , $a_{3}=2.1 b_{1}=11.25, b_{2}=7.95, b_{3}=4.2$

Question 44

Write a Fourier series for the function below:

Question 45

Write seven terms of the Fourier series given the following coefficients: $a_{0}=1, a_{1}=2, a_{2}=3, a_{3}=4$ , $b_{1}=1.5, b_{2}=2.5, b_{3}=3.5$

Question 46

Label the function below as odd, even, or neither.

Question 47

Label the function below as odd, even, or neither.

Question 48

Does the function below have half-wave symmetry?

Question 49

Does the function below have half-wave symmetry?

Question 50

Label the function below as odd, even, or neither.

Question 51

Label the function below as odd, even, or neither.

Question 52

Verify that the first four terms of the Fourier series for the full wave rectification of the sine function
$f(t)=\sin (2 \pi f t){f=1 \text { below are: }} f(t)=\frac{2}{\pi}\left[1-2\left(\frac{1}{3} \cos 2 \pi t+\frac{1}{15} \cos 4 \pi t+\frac{1}{35} \cos 6 \pi t\right)\right]$

Question 53

Verify that the first four terms of the Fourier series for the half wave Rectification of the sine function
$f(t)=\sin (2 \pi f t) {f=1} \text { below are: } f(t)=\frac{\sin \pi t}{2}+\frac{1}{\pi}\left[1-\frac{2}{3} \cos 4 \pi t+\frac{2}{15} \cos 8 \pi t\right]$

Question 54

Verify that the first four terms of the Fourier series for the sawtooth function below are:
$f(t)=\frac{1}{2}-\frac{1}{\pi}\left[\sin 2 \pi t+\frac{1}{2} \sin 4 \pi t+\frac{1}{3} \sin 6 \pi t\right]$

Question 55

Write a Fourier series for the waveform below:

Question 56

Find the first six terms of the following waveform. Assume half-wave symmetry.
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline\boldsymbol{x}& 0^{\circ} & 20^{\circ} & 40^{\circ} & 60^{\circ} & 80^{\circ} & 100^{\circ} & 120^{\circ} & 140^{\circ} & 160^{\circ} &180^{\circ} \\\hline\boldsymbol{y} & 0 & 1.9 & 4.0 & 5.1 & 6.8 & 9.7 & 15.2 & 15.5 & 12.9 & 0 \\\hline\end{array}$

Question 57

Find the first four terms (rounded to three decimal places) of the Fourier series. Assume half-wave symmetry.
$\begin{array}{|c|c|c|c|c|c|c|c|}\hline\boldsymbol{x} & 0^{\circ} & 30^{\circ} & 60^{\circ} & 90^{\circ} & 120^{\circ} & 160^{\circ} & 180^{\circ} \\\hline\boldsymbol{y} & 0 & 3.1 & 4.5 & 8.1 & 15.1 & 14.4 & 0 \\\hline\end{array}$

Question 58

Find the first four terms (rounded to three decimal places) of the Fourier series. Assume half-wave symmetry.
$\begin{array}{|c|c|c|c|c|c|c|c|}\hline\boldsymbol{x} & 0^{\circ} & 30^{\circ} & 60^{\circ} & 90^{\circ} & 120^{\circ} & 160^{\circ} & 180^{\circ} \\\hline\boldsymbol{y} & 0 & 6.8 & 7.4 & 9.1 & 8.3 & 5.2 & 0 \\\hline\end{array}$

Question 59

Find the first six terms (rounded to three decimal places) of the Fourier series. Assume half-wave symmetry.
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline\boldsymbol{x} & 0^{\circ} & 20^{\circ} & 40^{\circ} & 60^{\circ} & 80^{\circ} & 100^{\circ} & 120^{\circ} & 140^{\circ} & 160^{\circ} & 180^{\circ} \\\hline\boldsymbol{y} & 0 & 4 & 7.5 & 10 & 8 & 6.5 & 5 & 4 & 2 & 0 \\\hline\end{array}$

Question 60

Find the first six terms (rounded to three decimal places) of the Fourier series. Assume half-wave symmetry.
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline\boldsymbol{x} & 0^{\circ} & 20^{\circ} & 40^{\circ} & 60^{\circ} & 80^{\circ} & 100^{\circ} & 120^{\circ} & 140^{\circ} & 160^{\circ} & 180^{\circ} \\\hline\boldsymbol{y} & 0 & 2.1 & 4.8 & 7.5 & 11.2 & 10.9 & 6.8 & 5.2 & 2.4 & 0 \\\hline\end{array}$