## Problem Set: Working with Taylor Series

In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial.

1. ${\left(1-x\right)}^{\frac{1}{3}}$

2. ${\left(1+{x}^{2}\right)}^{\frac{-1}{3}}$

3. ${\left(1-x\right)}^{1.01}$

4. ${\left(1 - 2x\right)}^{\frac{2}{3}}$

In the following exercises, use the substitution ${\left(b+x\right)}^{r}={\left(b+a\right)}^{r}{\left(1+\frac{x-a}{b+a}\right)}^{r}$ in the binomial expansion to find the Taylor series of each function with the given center.

5. $\sqrt{x+2}$ at $a=0$

6. $\sqrt{{x}^{2}+2}$ at $a=0$

7. $\sqrt{x+2}$ at $a=1$

8. $\sqrt{2x-{x}^{2}}$ at $a=1$ (Hint: $2x-{x}^{2}=1-{\left(x - 1\right)}^{2}$)

9. ${\left(x - 8\right)}^{\frac{1}{3}}$ at $a=9$

10. $\sqrt{x}$ at $a=4$

11. ${x}^{\frac{1}{3}}$ at $a=27$

12. $\sqrt{x}$ at $x=9$

In the following exercises, use the binomial theorem to estimate each number, computing enough terms to obtain an estimate accurate to an error of at most $\frac{1}{1000}$.

13. [T] ${\left(15\right)}^{\frac{1}{4}}$ using ${\left(16-x\right)}^{\frac{1}{4}}$

14. [T] ${\left(1001\right)}^{\frac{1}{3}}$ using ${\left(1000+x\right)}^{\frac{1}{3}}$

In the following exercises, use the binomial approximation $\sqrt{1-x}\approx 1-\frac{x}{2}-\frac{{x}^{2}}{8}-\frac{{x}^{3}}{16}-\frac{5{x}^{4}}{128}-\frac{7{x}^{5}}{256}$ for $|x|<1$ to approximate each number. Compare this value to the value given by a scientific calculator.

15. [T] $\frac{1}{\sqrt{2}}$ using $x=\frac{1}{2}$ in ${\left(1-x\right)}^{\frac{1}{2}}$

16. [T] $\sqrt{5}=5\times \frac{1}{\sqrt{5}}$ using $x=\frac{4}{5}$ in ${\left(1-x\right)}^{\frac{1}{2}}$

17. [T] $\sqrt{3}=\frac{3}{\sqrt{3}}$ using $x=\frac{2}{3}$ in ${\left(1-x\right)}^{\frac{1}{2}}$

18. [T] $\sqrt{6}$ using $x=\frac{5}{6}$ in ${\left(1-x\right)}^{\frac{1}{2}}$

19. Integrate the binomial approximation of $\sqrt{1-x}$ to find an approximation of ${\displaystyle\int }_{0}^{x}\sqrt{1-t}dt$.

20. [T] Recall that the graph of $\sqrt{1-{x}^{2}}$ is an upper semicircle of radius $1$. Integrate the binomial approximation of $\sqrt{1-{x}^{2}}$ up to order $8$ from $x=-1$ to $x=1$ to estimate $\frac{\pi }{2}$.

In the following exercises, use the expansion ${\left(1+x\right)}^{\frac{1}{3}}=1+\frac{1}{3}x-\frac{1}{9}{x}^{2}+\frac{5}{81}{x}^{3}-\frac{10}{243}{x}^{4}+\cdots$ to write the first five terms (not necessarily a quartic polynomial) of each expression.

21. ${\left(1+4x\right)}^{\frac{1}{3}};a=0$

22. ${\left(1+4x\right)}^{\frac{4}{3}};a=0$

23. ${\left(3+2x\right)}^{\frac{1}{3}};a=-1$

24. ${\left({x}^{2}+6x+10\right)}^{\frac{1}{3}};a=-3$

25. Use ${\left(1+x\right)}^{\frac{1}{3}}=1+\frac{1}{3}x-\frac{1}{9}{x}^{2}+\frac{5}{81}{x}^{3}-\frac{10}{243}{x}^{4}+\cdots$ with $x=1$ to approximate ${2}^{\frac{1}{3}}$.

26. Use the approximation ${\left(1-x\right)}^{\frac{2}{3}}=1-\frac{2x}{3}-\frac{{x}^{2}}{9}-\frac{4{x}^{3}}{81}-\frac{7{x}^{4}}{243}-\frac{14{x}^{5}}{729}+\cdots$ for $|x|<1$ to approximate ${2}^{\frac{1}{3}}={2.2}^{\frac{-2}{3}}$.

27. Find the $25\text{th}$ derivative of $f\left(x\right)={\left(1+{x}^{2}\right)}^{13}$ at $x=0$.

28. Find the $99$ th derivative of $f\left(x\right)={\left(1+{x}^{4}\right)}^{25}$.

In the following exercises, find the Maclaurin series of each function.

29. $f\left(x\right)=x{e}^{2x}$

30. $f\left(x\right)={2}^{x}$

31. $f\left(x\right)=\frac{\sin{x}}{x}$

32. $f\left(x\right)=\frac{\sin\left(\sqrt{x}\right)}{\sqrt{x}},\left(x>0\right)$,

33. $f\left(x\right)=\sin\left({x}^{2}\right)$

34. $f\left(x\right)={e}^{{x}^{3}}$

35. $f\left(x\right)={\cos}^{2}x$ using the identity ${\cos}^{2}x=\frac{1}{2}+\frac{1}{2}\cos\left(2x\right)$

36. $f\left(x\right)={\sin}^{2}x$ using the identity ${\sin}^{2}x=\frac{1}{2}-\frac{1}{2}\cos\left(2x\right)$

In the following exercises, find the Maclaurin series of $F\left(x\right)={\displaystyle\int }_{0}^{x}f\left(t\right)dt$ by integrating the Maclaurin series of $f$ term by term. If $f$ is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero.

37. $F\left(x\right)={\displaystyle\int }_{0}^{x}{e}^{\text{-}{t}^{2}}dt;f\left(t\right)={e}^{\text{-}{t}^{2}}=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{2n}}{n\text{!}}$

38. $F\left(x\right)={\tan}^{-1}x;f\left(t\right)=\frac{1}{1+{t}^{2}}=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}{t}^{2n}$

39. $F\left(x\right)={\text{tanh}}^{-1}x;f\left(t\right)=\frac{1}{1-{t}^{2}}=\displaystyle\sum _{n=0}^{\infty }{t}^{2n}$

40. $F\left(x\right)={\sin}^{-1}x;f\left(t\right)=\frac{1}{\sqrt{1-{t}^{2}}}=\displaystyle\sum _{k=0}^{\infty }\left(\begin{array}{c}\frac{1}{2}\hfill \\ k\hfill \end{array}\right)\frac{{t}^{2k}}{k\text{!}}$

41. $F\left(x\right)={\displaystyle\int }_{0}^{x}\frac{\sin{t}}{t}dt;f\left(t\right)=\frac{\sin{t}}{t}=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{2n}}{\left(2n+1\right)\text{!}}$

42. $F\left(x\right)={\displaystyle\int }_{0}^{x}\cos\left(\sqrt{t}\right)dt;f\left(t\right)=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{n}}{\left(2n\right)\text{!}}$

43. $F\left(x\right)={\displaystyle\int }_{0}^{x}\frac{1-\cos{t}}{{t}^{2}}dt;f\left(t\right)=\frac{1-\cos{t}}{{t}^{2}}=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{2n}}{\left(2n+2\right)\text{!}}$

44. $F\left(x\right)={\displaystyle\int }_{0}^{x}\frac{\text{ln}\left(1+t\right)}{t}dt;f\left(t\right)=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{n}}{n+1}$

In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f$.

45. $f\left(x\right)=\sin\left(x+\frac{\pi }{4}\right)=\sin{x}\cos\left(\frac{\pi }{4}\right)+\cos{x}\sin\left(\frac{\pi }{4}\right)$

46. $f\left(x\right)=\tan{x}$

47. $f\left(x\right)=\text{ln}\left(\cos{x}\right)$

48. $f\left(x\right)={e}^{x}\cos{x}$

49. $f\left(x\right)={e}^{\sin{x}}$

50. $f\left(x\right)={\sec}^{2}x$

51. $f\left(x\right)=\text{tanh}x$

52. $f\left(x\right)=\frac{\tan\sqrt{x}}{\sqrt{x}}$ (see expansion for $\tan{x}$)

In the following exercises, find the radius of convergence of the Maclaurin series of each function.

53. $\text{ln}\left(1+x\right)$

54. $\frac{1}{1+{x}^{2}}$

55. ${\tan}^{-1}x$

56. $\text{ln}\left(1+{x}^{2}\right)$

57. Find the Maclaurin series of $\text{sinh}x=\frac{{e}^{x}-{e}^{\text{-}x}}{2}$.

58. Find the Maclaurin series of $\text{cosh}x=\frac{{e}^{x}+{e}^{\text{-}x}}{2}$.

59. Differentiate term by term the Maclaurin series of $\text{sinh}x$ and compare the result with the Maclaurin series of $\text{cosh}x$.

60. [T] Let ${S}_{n}\left(x\right)=\displaystyle\sum _{k=0}^{n}{\left(-1\right)}^{k}\frac{{x}^{2k+1}}{\left(2k+1\right)\text{!}}$ and ${C}_{n}\left(x\right)=\displaystyle\sum _{n=0}^{n}{\left(-1\right)}^{k}\frac{{x}^{2k}}{\left(2k\right)\text{!}}$ denote the respective Maclaurin polynomials of degree $2n+1$ of $\sin{x}$ and degree $2n$ of $\cos{x}$. Plot the errors $\frac{{S}_{n}\left(x\right)}{{C}_{n}\left(x\right)}-\tan{x}$ for $n=1,..,5$ and compare them to $x+\frac{{x}^{3}}{3}+\frac{2{x}^{5}}{15}+\frac{17{x}^{7}}{315}-\tan{x}$ on $\left(-\frac{\pi }{4},\frac{\pi }{4}\right)$.

61. Use the identity $2\sin{x}\cos{x}=\sin\left(2x\right)$ to find the power series expansion of ${\sin}^{2}x$ at $x=0$. (Hint: Integrate the Maclaurin series of $\sin\left(2x\right)$ term by term.)

62. If $y=\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$, find the power series expansions of $x{y}^{\prime }$ and ${x}^{2}y\text{''}$.

63. [T] Suppose that $y=\displaystyle\sum _{k=0}^{\infty }{a}_{k}{x}^{k}$ satisfies ${y}^{\prime }=-2xy$ and $y\left(0\right)=0$. Show that ${a}_{2k+1}=0$ for all $k$ and that ${a}_{2k+2}=\frac{\text{-}{a}_{2k}}{k+1}$. Plot the partial sum ${S}_{20}$ of $y$ on the interval $\left[-4,4\right]$.

64. [T] Suppose that a set of standardized test scores is normally distributed with mean $\mu =100$ and standard deviation $\sigma =10$. Set up an integral that represents the probability that a test score will be between $90$ and $110$ and use the integral of the degree $10$ Maclaurin polynomial of $\frac{1}{\sqrt{2\pi }}{e}^{\frac{\text{-}{x}^{2}}{2}}$ to estimate this probability.

65. [T] Suppose that a set of standardized test scores is normally distributed with mean $\mu =100$ and standard deviation $\sigma =10$. Set up an integral that represents the probability that a test score will be between $70$ and $130$ and use the integral of the degree $50$ Maclaurin polynomial of $\frac{1}{\sqrt{2\pi }}{e}^{\frac{\text{-}{x}^{2}}{2}}$ to estimate this probability.

66. [T] Suppose that $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $f\left(x\right)$ such that $f\left(0\right)=1,{f}^{\prime }\left(0\right)=0$, and $f\text{''}\left(x\right)=\text{-}f\left(x\right)$. Find a formula for ${a}_{n}$ and plot the partial sum ${S}_{N}$ for $N=20$ on $\left[-5,5\right]$.

67. [T] Suppose that $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $f\left(x\right)$ such that $f\left(0\right)=0,{f}^{\prime }\left(0\right)=1$, and $f\text{''}\left(x\right)=\text{-}f\left(x\right)$. Find a formula for ${a}_{n}$ and plot the partial sum ${S}_{N}$ for $N=10$ on $\left[-5,5\right]$.

68. Suppose that $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $y$ such that $y\text{''}-{y}^{\prime }+y=0$ where $y\left(0\right)=1$ and $y^{\prime} \left(0\right)=0$. Find a formula that relates ${a}_{n+2},{a}_{n+1}$, and ${a}_{n}$ and compute ${a}_{0},…,{a}_{5}$.

69. Suppose that $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $y$ such that $y\text{''}-{y}^{\prime }+y=0$ where $y\left(0\right)=0$ and ${y}^{\prime }\left(0\right)=1$. Find a formula that relates ${a}_{n+2},{a}_{n+1}$, and ${a}_{n}$ and compute ${a}_{1},…,{a}_{5}$.

The error in approximating the integral ${\displaystyle\int }_{a}^{b}f\left(t\right)dt$ by that of a Taylor approximation ${\displaystyle\int }_{a}^{b}{P}_{n}\left(t\right)dt$ is at most ${\displaystyle\int }_{a}^{b}{R}_{n}\left(t\right)dt$. In the following exercises, the Taylor remainder estimate ${R}_{n}\le \frac{M}{\left(n+1\right)\text{!}}{|x-a|}^{n+1}$ guarantees that the integral of the Taylor polynomial of the given order approximates the integral of $f$ with an error less than $\frac{1}{10}$.

1. Evaluate the integral of the appropriate Taylor polynomial and verify that it approximates the CAS value with an error less than $\frac{1}{100}$.
2. Compare the accuracy of the polynomial integral estimate with the remainder estimate.

70. [T] ${\displaystyle\int }_{0}^{\pi }\frac{\sin{t}}{t}dt;{P}_{s}=1-\frac{{x}^{2}}{3\text{!}}+\frac{{x}^{4}}{5\text{!}}-\frac{{x}^{6}}{7\text{!}}+\frac{{x}^{8}}{9\text{!}}$ (You may assume that the absolute value of the ninth derivative of $\frac{\sin{t}}{t}$ is bounded by $0.1.$)

71. [T] ${\displaystyle\int }_{0}^{2}{e}^{\text{-}{x}^{2}}dx;{p}_{11}=1-{x}^{2}+\frac{{x}^{4}}{2}-\frac{{x}^{6}}{3\text{!}}+\cdots-\frac{{x}^{22}}{11\text{!}}$ (You may assume that the absolute value of the $23\text{rd}$ derivative of ${e}^{\text{-}{x}^{2}}$ is less than $2\times {10}^{14}.$)

The following exercises deal with Fresnel integrals.

72. The Fresnel integrals are defined by $C\left(x\right)={\displaystyle\int }_{0}^{x}\cos\left({t}^{2}\right)dt$ and $S\left(x\right)={\displaystyle\int }_{0}^{x}\sin\left({t}^{2}\right)dt$. Compute the power series of $C\left(x\right)$ and $S\left(x\right)$ and plot the sums ${C}_{N}\left(x\right)$ and ${S}_{N}\left(x\right)$ of the first $N=50$ nonzero terms on $\left[0,2\pi \right]$.

73. [T] The Fresnel integrals are used in design applications for roadways and railways and other applications because of the curvature properties of the curve with coordinates $\left(C\left(t\right),S\left(t\right)\right)$. Plot the curve $\left({C}_{50},{S}_{50}\right)$ for $0\le t\le 2\pi$, the coordinates of which were computed in the previous exercise.

74. Estimate ${\displaystyle\int }_{0}^{\frac{1}{4}}\sqrt{x-{x}^{2}}dx$ by approximating $\sqrt{1-x}$ using the binomial approximation $1-\frac{x}{2}-\frac{{x}^{2}}{8}-\frac{{x}^{3}}{16}-\frac{5{x}^{4}}{2128}-\frac{7{x}^{5}}{256}$.

75. [T] Use Newton’s approximation of the binomial $\sqrt{1-{x}^{2}}$ to approximate $\pi$ as follows. The circle centered at $\left(\frac{1}{2},0\right)$ with radius $\frac{1}{2}$ has upper semicircle $y=\sqrt{x}\sqrt{1-x}$. The sector of this circle bounded by the $x$ -axis between $x=0$ and $x=\frac{1}{2}$ and by the line joining $\left(\frac{1}{4},\frac{\sqrt{3}}{4}\right)$ corresponds to $\frac{1}{6}$ of the circle and has area $\frac{\pi }{24}$. This sector is the union of a right triangle with height $\frac{\sqrt{3}}{4}$ and base $\frac{1}{4}$ and the region below the graph between $x=0$ and $x=\frac{1}{4}$. To find the area of this region you can write $y=\sqrt{x}\sqrt{1-x}=\sqrt{x}\times \left(\text{binomial expansion of}\sqrt{1-x}\right)$ and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate $\pi$.

76. Use the approximation $T\approx 2\pi \sqrt{\frac{L}{g}}\left(1+\frac{{k}^{2}}{4}\right)$ to approximate the period of a pendulum having length $10$ meters and maximum angle ${\theta }_{\text{max}}=\frac{\pi }{6}$ where $k=\sin\left(\frac{{\theta }_{\text{max}}}{2}\right)$. Compare this with the small angle estimate $T\approx 2\pi \sqrt{\frac{L}{g}}$.

77. Suppose that a pendulum is to have a period of $2$ seconds and a maximum angle of ${\theta }_{\text{max}}=\frac{\pi }{6}$. Use $T\approx 2\pi \sqrt{\frac{L}{g}}\left(1+\frac{{k}^{2}}{4}\right)$ to approximate the desired length of the pendulum. What length is predicted by the small angle estimate $T\approx 2\pi \sqrt{\frac{L}{g}}?$

78. Evaluate ${\displaystyle\int }_{0}^{\frac{\pi}{2}}{\sin}^{4}\theta d\theta$ in the approximation $T=4\sqrt{\frac{L}{g}}{\displaystyle\int }_{0}^{\frac{\pi}{2}}\left(1+\frac{1}{2}{k}^{2}{\sin}^{2}\theta +\frac{3}{8}{k}^{4}{\sin}^{4}\theta +\cdots\right)d\theta$ to obtain an improved estimate for $T$.

79. [T] An equivalent formula for the period of a pendulum with amplitude ${\theta }_{\text{max}}$ is $T\left({\theta }_{\text{max}}\right)=2\sqrt{2}\sqrt{\frac{L}{g}}{\displaystyle\int }_{0}^{{\theta }_{\text{max}}}\frac{d\theta }{\sqrt{\cos\theta }-\cos\left({\theta }_{\text{max}}\right)}$ where $L$ is the pendulum length and $g$ is the gravitational acceleration constant. When ${\theta }_{\text{max}}=\frac{\pi }{3}$ we get $\frac{1}{\sqrt{\cos{t} - \frac{1}{2}}}\approx \sqrt{2}\left(1+\frac{{t}^{2}}{2}+\frac{{t}^{4}}{3}+\frac{181{t}^{6}}{720}\right)$. Integrate this approximation to estimate $T\left(\frac{\pi }{3}\right)$ in terms of $L$ and $g$. Assuming $g=9.806$ meters per second squared, find an approximate length $L$ such that $T\left(\frac{\pi }{3}\right)=2$ seconds.