Introduction to Power Series

In the following exercises (1-3), state whether each statement is true, or give an example to show that it is false.

1. If [latex]\displaystyle\sum {n=1}^{\infty }{a}{n}{x}^{n}[/latex] converges, then [latex]{a}_{n}{x}^{n}\to 0[/latex] as [latex]n\to \infty[/latex].
2. Given any sequence [latex]{a}_{n}[/latex], there is always some [latex]R>0[/latex], possibly very small, such that [latex]\displaystyle\sum {n=1}^{\infty }{a}{n}{x}^{n}[/latex] converges on [latex]\left(\text{-}R,R\right)[/latex].
3. Suppose that [latex]\displaystyle\sum {n=0}^{\infty }{a}{n}{\left(x - 3\right)}^{n}[/latex] converges at [latex]x=6[/latex]. At which of the following points must the series also converge? Use the fact that if [latex]\displaystyle\sum {a}_{n}{\left(x-c\right)}^{n}[/latex] converges at [latex]x[/latex], then it converges at any point closer to [latex]c[/latex] than [latex]x[/latex].
   1. [latex]x=1[/latex]
   2. [latex]x=2[/latex]
   3. [latex]x=3[/latex]
   4. [latex]x=0[/latex]
   5. [latex]x=5.99[/latex]
   6. [latex]x=0.000001[/latex]

In the following exercises (4-6), suppose that [latex]|\dfrac{{a}{n+1}}{{a}{n}}|\to 1[/latex] as [latex]n\to \infty[/latex]. Find the radius of convergence for each series.

4. [latex]\displaystyle\sum {n=0}^{\infty }{a}{n}{2}^{n}{x}^{n}[/latex]
5. [latex]\displaystyle\sum {n=0}^{\infty }\dfrac{{a}{n}{\pi }^{n}{x}^{n}}{{e}^{n}}[/latex]
6. [latex]\displaystyle\sum {n=0}^{\infty }{a}{n}{\left(-1\right)}^{n}{x}^{2n}[/latex]

In the following exercises (7-11), find the radius of convergence R and interval of convergence for [latex]\displaystyle\sum {a}{n}{x}^{n}[/latex] with the given coefficients [latex]{a}{n}[/latex].

7. [latex]\displaystyle\sum _{n=1}^{\infty }\dfrac{{\left(2x\right)}^{n}}{n}[/latex]
8. [latex]\displaystyle\sum _{n=1}^{\infty }\dfrac{n{x}^{n}}{{2}^{n}}[/latex]
9. [latex]\displaystyle\sum _{n=1}^{\infty }\dfrac{{n}^{2}{x}^{n}}{{2}^{n}}[/latex]
10. [latex]\displaystyle\sum _{k=1}^{\infty }\dfrac{{\pi }^{k}{x}^{k}}{{k}^{\pi }}[/latex]
11. [latex]\displaystyle\sum _{n=1}^{\infty }\dfrac{{10}^{n}{x}^{n}}{n\text{!}}[/latex]

In the following exercises (12-14), find the radius of convergence of each series.

12. [latex]\displaystyle\sum _{k=1}^{\infty }\dfrac{{\left(k\text{!}\right)}^{2}{x}^{k}}{\left(2k\right)\text{!}}[/latex]
13. [latex]\displaystyle\sum _{k=1}^{\infty }\dfrac{k\text{!}}{1\cdot 3\cdot 5\text{...}\left(2k - 1\right)}{x}^{k}[/latex]
14. [latex]\displaystyle\sum _{n=1}^{\infty }\dfrac{{x}^{n}}{\left(\begin{array}{c}2n\ n\end{array}\right)}[/latex] where [latex]\left(\begin{array}{c}n\ k\end{array}\right)=\dfrac{n\text{!}}{k\text{!}\left(n-k\right)\text{!}}[/latex]

In the following exercises (15-16), use the ratio test to determine the radius of convergence of each series.

15. [latex]\displaystyle\sum _{n=1}^{\infty }\dfrac{{\left(n\text{!}\right)}^{3}}{\left(3n\right)\text{!}}{x}^{n}[/latex]
16. [latex]\displaystyle\sum _{n=1}^{\infty }\dfrac{n\text{!}}{{n}^{n}}{x}^{n}[/latex]

In the following exercises (17-21), given that [latex]\dfrac{1}{1-x}=\displaystyle\sum _{n=0}^{\infty }{x}^{n}[/latex] with convergence in [latex]\left(-1,1\right)[/latex], find the power series for each function with the given center [latex]a[/latex], and identify its interval of convergence.

17. [latex]f\left(x\right)=\dfrac{1}{x};a=1[/latex] (Hint: [latex]\dfrac{1}{x}=\dfrac{1}{1-\left(1-x\right)}[/latex])
18. [latex]f\left(x\right)=\dfrac{x}{1-{x}^{2}};a=0[/latex]
19. [latex]f\left(x\right)=\dfrac{{x}^{2}}{1+{x}^{2}};a=0[/latex]
20. [latex]f\left(x\right)=\dfrac{1}{1 - 2x};a=0[/latex].
21. [latex]f\left(x\right)=\dfrac{{x}^{2}}{1 - 4{x}^{2}};a=0[/latex]

In the following exercises (22-23), suppose that [latex]p\left(x\right)=\displaystyle\sum {n=0}^{\infty }{a}{n}{x}^{n}[/latex] satisfies [latex]\underset{n\to \infty }{\text{lim}}\dfrac{{a}{n+1}}{{a}{n}}=1[/latex] where [latex]{a}_{n}\ge 0[/latex] for each [latex]n[/latex]. State whether each series converges on the full interval [latex]\left(-1,1\right)[/latex], or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.

22. [latex]\displaystyle\sum {n=0}^{\infty }{a}{2n}{x}^{2n}[/latex]
23. [latex]\displaystyle\sum {n=0}^{\infty }{a}{{n}^{2}}{x}^{{n}^{2}}[/latex] (Hint: Let [latex]{b}{k}={a}{k}[/latex] if [latex]k={n}^{2}[/latex] for some n, otherwise [latex]{b}_{k}=0.[/latex])

For the following exercises (24-25), solve each problem.

24. Plot the graphs of [latex]\dfrac{1}{1-x}[/latex] and of the partial sums [latex]{S}_{N}=\displaystyle\sum {n=0}^{N}{x}^{n}[/latex] for [latex]n=10,20,30[/latex] on the interval [latex]\left[-0.99,0.99\right][/latex]. Comment on the approximation of [latex]\dfrac{1}{1-x}[/latex] by [latex]{S}{N}[/latex] near [latex]x=-1[/latex] and near [latex]x=1[/latex] as N increases.
25. Plot the graphs of the partial sums [latex]{S}_{n}=\displaystyle\sum _{n=1}^{N}\dfrac{{x}^{n}}{{n}^{2}}[/latex] for [latex]n=10,50,100[/latex] on the interval [latex]\left[-0.99,0.99\right][/latex]. Comment on the behavior of the sums near [latex]x=-1[/latex] and near [latex]x=1[/latex] as N increases.

Operations with Power Series

1. If [latex]f\left(x\right)=\displaystyle\sum _{n=0}^{\infty }\dfrac{{x}^{n}}{n\text{!}}[/latex] and [latex]g\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\dfrac{{x}^{n}}{n\text{!}}[/latex], find the power series of [latex]\dfrac{1}{2}\left(f\left(x\right)+g\left(x\right)\right)[/latex] and of [latex]\dfrac{1}{2}\left(f\left(x\right)-g\left(x\right)\right)[/latex].

In the following exercises (2-3), use partial fractions to find the power series of each function.

2. [latex]\dfrac{4}{\left(x - 3\right)\left(x+1\right)}[/latex]
3. [latex]\dfrac{5}{\left({x}^{2}+4\right)\left({x}^{2}-1\right)}[/latex]

In the following exercises (4-6), express each series as a rational function.

4. [latex]\displaystyle\sum _{n=1}^{\infty }\dfrac{1}{{x}^{n}}[/latex]
5. [latex]\displaystyle\sum _{n=1}^{\infty }\dfrac{1}{{\left(x - 3\right)}^{2n - 1}}[/latex]

The following exercises (6-8) explore applications of annuities.

6. Calculate the present values [latex]P[/latex] of an annuity in which [latex]$10,000[/latex] is to be paid out annually for a period of [latex]20[/latex] years, assuming interest rates of [latex]r=0.03,r=0.05[/latex], and [latex]r=0.07[/latex].
7. Calculate the annual payouts [latex]C[/latex] to be given for [latex]20[/latex] years on annuities having present value [latex]$100,000[/latex] assuming respective interest rates of [latex]r=0.03,r=0.05[/latex], and [latex]r=0.07[/latex].
8. Suppose that an annuity has a present value [latex]P=1\text{million dollars}[/latex]. What interest rate [latex]r[/latex] would allow for perpetual annual payouts of [latex]$50,000[/latex]?

In the following exercises (9-10), express the sum of each power series in terms of geometric series, and then express the sum as a rational function.

9. [latex]x+{x}^{2}-{x}^{3}+{x}^{4}+{x}^{5}-{x}^{6}+\cdots[/latex]
10. [latex]x-{x}^{2}-{x}^{3}+{x}^{4}-{x}^{5}-{x}^{6}+{x}^{7}-\cdots[/latex]

In the following exercises (11-12), find the power series of [latex]f\left(x\right)g\left(x\right)[/latex] given [latex]f[/latex] and [latex]g[/latex] as defined.

11. [latex]f\left(x\right)=2\displaystyle\sum _{n=0}^{\infty }{x}^{n},g\left(x\right)=\displaystyle\sum _{n=0}^{\infty }n{x}^{n}[/latex]
12. [latex]f\left(x\right)=g\left(x\right)=\displaystyle\sum _{n=1}^{\infty }{\left(\dfrac{x}{2}\right)}^{n}[/latex]

In the following exercise, differentiate the given series expansion of [latex]f[/latex] term-by-term to obtain the corresponding series expansion for the derivative of [latex]f[/latex].

13. [latex]f\left(x\right)=\dfrac{1}{1+x}=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}{x}^{n}[/latex]

In the following exercise, integrate the given series expansion of [latex]f[/latex] term-by-term from zero to [latex]x[/latex] to obtain the corresponding series expansion for the indefinite integral of [latex]f[/latex].

14. [latex]f\left(x\right)=\dfrac{2x}{{\left(1+{x}^{2}\right)}^{2}}=\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}\left(2n\right){x}^{2n - 1}[/latex]

In the following exercises (15-16), evaluate each infinite series by identifying it as the value of a derivative or integral of geometric series.

15. Evaluate [latex]\displaystyle\sum _{n=1}^{\infty }\dfrac{n}{{2}^{n}}[/latex] as [latex]{f}^{\prime }\left(\dfrac{1}{2}\right)[/latex] where [latex]f\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{x}^{n}[/latex].
16. Evaluate [latex]\displaystyle\sum _{n=2}^{\infty }\dfrac{n\left(n - 1\right)}{{2}^{n}}[/latex] as [latex]f''\left(\dfrac{1}{2}\right)[/latex] where [latex]f\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{x}^{n}[/latex].

In the following exercises (17-20), given that [latex]\dfrac{1}{1-x}=\displaystyle\sum _{n=0}^{\infty }{x}^{n}[/latex], use term-by-term differentiation or integration to find power series for each function centered at the given point.

17. [latex]f\left(x\right)=\text{ln}x[/latex] centered at [latex]x=1[/latex] (Hint: [latex]x=1-\left(1-x\right)[/latex])
18. [latex]\text{ln}\left(1-{x}^{2}\right)[/latex] at [latex]x=0[/latex]
19. [latex]f\left(x\right)={\tan}^{-1}\left({x}^{2}\right)[/latex] at [latex]x=0[/latex]
20. [latex]f\left(x\right)={\displaystyle\int }_{0}^{x}\text{ln}tdt[/latex] where [latex]\text{ln}\left(x\right)=\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n - 1}\dfrac{{\left(x - 1\right)}^{n}}{n}[/latex]

In the following exercises (21-22), using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum.

21. [latex]\displaystyle\sum _{k=1}^{\infty }\dfrac{{x}^{3k}}{6k}[/latex]
22. [latex]\displaystyle\sum _{k=1}^{\infty }{2}^{\text{-}kx}[/latex] using [latex]y={2}^{\text{-}x}[/latex]

For the following exercises (23-26), solve each problem.

23. Differentiate the series [latex]E\left(x\right)=\displaystyle\sum _{n=0}^{\infty }\dfrac{{x}^{n}}{n\text{!}}[/latex] term-by-term to show that [latex]E\left(x\right)[/latex] is equal to its derivative.
24. Suppose that the coefficients an of the series [latex]\displaystyle\sum {n=0}^{\infty }{a}{n}{x}^{n}[/latex] are defined by the recurrence relation [latex]{a}{n}=\dfrac{{a}{n - 1}}{n}+\dfrac{{a}{n - 2}}{n\left(n - 1\right)}[/latex]. For [latex]{a}{0}=0[/latex] and [latex]{a}{1}=1[/latex], compute and plot the sums [latex]{S}{N}=\displaystyle\sum {n=0}^{N}{a}{n}{x}^{n}[/latex] for [latex]N=2,3,4,5[/latex] on [latex]\left[-1,1\right][/latex].
25. Given the power series expansion [latex]\text{ln}\left(1+x\right)=\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n - 1}\dfrac{{x}^{n}}{n}[/latex], determine how many terms N of the sum evaluated at [latex]x=-\dfrac{1}{2}[/latex] are needed to approximate [latex]\text{ln}\left(2\right)[/latex] accurate to within [latex]\dfrac{1}{1000}[/latex]. Evaluate the corresponding partial sum [latex]\displaystyle\sum _{n=1}^{N}{\left(-1\right)}^{n - 1}\dfrac{{x}^{n}}{n}[/latex].
26. Recall that [latex]{\tan}^{-1}\left(\dfrac{1}{\sqrt{3}}\right)=\dfrac{\pi }{6}[/latex]. Assuming an exact value of [latex]\left(\dfrac{1}{\sqrt{3}}\right)[/latex], estimate [latex]\dfrac{\pi }{6}[/latex] by evaluating partial sums [latex]{S}_{N}\left(\dfrac{1}{\sqrt{3}}\right)[/latex] of the power series expansion [latex]{\tan}^{-1}\left(x\right)=\displaystyle\sum {k=0}^{\infty }{\left(-1\right)}^{k}\dfrac{{x}^{2k+1}}{2k+1}[/latex] at [latex]x=\dfrac{1}{\sqrt{3}}[/latex]. What is the smallest number N such that [latex]6{S}{N}\left(\dfrac{1}{\sqrt{3}}\right)[/latex] approximates π accurately to within [latex]0.001[/latex]? How many terms are needed for accuracy to within [latex]0.00001[/latex]?

Taylor and Maclaurin Series

In the following exercises (1-4), find the Taylor polynomials of degree two approximating the given function centered at the given point.

1. [latex]f\left(x\right)=1+x+{x}^{2}[/latex] at [latex]a=-1[/latex]
2. [latex]f\left(x\right)=\sin\left(2x\right)[/latex] at [latex]a=\dfrac{\pi }{2}[/latex]
3. [latex]f\left(x\right)=\text{ln}x[/latex] at [latex]a=1[/latex]
4. [latex]f\left(x\right)={e}^{x}[/latex] at [latex]a=1[/latex]

In the following exercises (5-7), verify that the given choice of n in the remainder estimate [latex]|{R}{n}|\le \dfrac{M}{\left(n+1\right)\text{!}}{\left(x-a\right)}^{n+1}[/latex], where [latex]M[/latex] is the maximum value of [latex]|{f}^{\left(n+1\right)}\left(z\right)|[/latex] on the interval between a and the indicated point, yields [latex]|{R}{n}|\le \dfrac{1}{1000}[/latex]. Find the value of the Taylor polynomial [latex]p_n[/latex] of [latex]f[/latex] at the indicated point.

5. [latex]{\left(28\right)}^{\dfrac{1}{3}};a=27,n=1[/latex]
6. [latex]e^2[/latex]; [latex]a=0,n=9[/latex]
7. [latex]\text{ln}\left(2\right);a=1,n=1000[/latex]
8. Integrate the approximation [latex]{e}^{x}\approx 1+x+\dfrac{{x}^{2}}{2}+\cdots+\dfrac{{x}^{6}}{720}[/latex] evaluated at −x2 to approximate [latex]{\displaystyle\int }_{0}^{1}{e}^{\text{-}{x}^{2}}dx[/latex].

In the following exercises (9-10), find the smallest value of n such that the remainder estimate [latex]|{R}{n}|\le \dfrac{M}{\left(n+1\right)\text{!}}{\left(x-a\right)}^{n+1}[/latex], where [latex]M[/latex] is the maximum value of [latex]|{f}^{\left(n+1\right)}\left(z\right)|[/latex] on the interval between a and the indicated point, yields [latex]|{R}{n}|\le \dfrac{1}{1000}[/latex] on the indicated interval.

9. [latex]f\left(x\right)=\cos{x}[/latex] on [latex]\left[-\dfrac{\pi }{2},\dfrac{\pi }{2}\right],a=0[/latex]
10. [latex]f\left(x\right)={e}^{\text{-}x}[/latex] on [latex]\left[-3,3\right],a=0[/latex]

In the following exercises (11-12), the maximum of the right-hand side of the remainder estimate [latex]|{R}_{1}|\le \dfrac{\text{max}|f\text{''}\left(z\right)|}{2}{R}^{2}[/latex] on [latex]\left[a-R,a+R\right][/latex] occurs at a or [latex]a\pm R[/latex]. Estimate the maximum value of [latex]R[/latex] such that [latex]\dfrac{\text{max}|f\text{''}\left(z\right)|}{2}{R}^{2}\le 0.1[/latex] on [latex]\left[a-R,a+R\right][/latex] by plotting this maximum as a function of [latex]R[/latex].

11. [latex]\sin{x}[/latex] approximated by x, [latex]a=0[/latex]
12. [latex]\cos{x}[/latex] approximated by [latex]1,a=0[/latex]

In the following exercises (13-17), find the Taylor series of the given function centered at the indicated point.

13. [latex]1+x+{x}^{2}+{x}^{3}[/latex] at [latex]a=-1[/latex]
14. [latex]\cos{x}[/latex] at [latex]a=2\pi[/latex]
15. [latex]\cos{x}[/latex] at [latex]x=\dfrac{\pi }{2}[/latex]
16. [latex]{e}^{x}[/latex] at [latex]a=1[/latex]
17. [latex]\dfrac{1}{{\left(x - 1\right)}^{3}}[/latex] at [latex]a=0[/latex]

In the following exercises (18-22), compute the Taylor series of each function around [latex]x=1[/latex].

18. [latex]f\left(x\right)=2-x[/latex]
19. [latex]f\left(x\right)={\left(x - 2\right)}^{2}[/latex]
20. [latex]f\left(x\right)=\dfrac{1}{x}[/latex]
21. [latex]f\left(x\right)=\dfrac{x}{4x - 2{x}^{2}-1}[/latex]
22. [latex]f\left(x\right)={e}^{2x}[/latex]

In the following exercises (23-24), identify the value of [latex]x[/latex] such that the given series [latex]\displaystyle\sum {n=0}^{\infty }{a}{n}[/latex] is the value of the Maclaurin series of [latex]f\left(x\right)[/latex] at [latex]x[/latex]. Approximate the value of [latex]f\left(x\right)[/latex] using [latex]{S}_{10}=\displaystyle\sum {n=0}^{10}{a}{n}[/latex].

23. [latex]\displaystyle\sum _{n=0}^{\infty }\dfrac{{2}^{n}}{n\text{!}}[/latex]
24. [latex]\displaystyle\sum _{n=0}^{\infty }\dfrac{{\left(-1\right)}^{n}{\left(2\pi \right)}^{2n+1}}{\left(2n+1\right)\text{!}}[/latex]

The following exercises (25-26) make use of the functions [latex]{S}{5}\left(x\right)=x-\dfrac{{x}^{3}}{6}+\dfrac{{x}^{5}}{120}[/latex] and [latex]{C}{4}\left(x\right)=1-\dfrac{{x}^{2}}{2}+\dfrac{{x}^{4}}{24}[/latex] on [latex]\left[\text{-}\pi ,\pi \right][/latex].

25. Plot [latex]{\cos}^{2}x-{\left({C}_{4}\left(x\right)\right)}^{2}[/latex] on [latex]\left[\text{-}\pi ,\pi \right][/latex]. Compare the maximum difference with the square of the Taylor remainder estimate for [latex]\cos{x}[/latex].
26. Compare [latex]\dfrac{{S}{5}\left(x\right)}{{C}{4}\left(x\right)}[/latex] on [latex]\left[-1,1\right][/latex] to [latex]\tan{x}[/latex]. Compare this with the Taylor remainder estimate for the approximation of [latex]\tan{x}[/latex] by [latex]x+\dfrac{{x}^{3}}{3}+\dfrac{2{x}^{5}}{15}[/latex].

In the following exercises (27-28), use the fact that if [latex]q\left(x\right)=\displaystyle\sum {n=1}^{\infty }{a}{n}{\left(x-c\right)}^{n}[/latex] converges in an interval containing [latex]c[/latex], then [latex]\underset{x\to c}{\text{lim}}q\left(x\right)={a}_{0}^{}[/latex] to evaluate each limit using Taylor series.

27. [latex]\underset{x\to 0}{\text{lim}}\dfrac{\text{ln}\left(1-{x}^{2}\right)}{{x}^{2}}[/latex]
28. [latex]\underset{x\to {0}^{+}}{\text{lim}}\dfrac{\cos\left(\sqrt{x}\right)-1}{2x}[/latex]

Applications of Series

In the following exercises (1-2), use appropriate substitutions to write down the Maclaurin series for the given binomial.

1. [latex]{\left(1+{x}^{2}\right)}^{\dfrac{-1}{3}}[/latex]
2. [latex]{\left(1 - 2x\right)}^{\dfrac{2}{3}}[/latex]

In the following exercises (3-6), use the substitution [latex]{\left(b+x\right)}^{r}={\left(b+a\right)}^{r}{\left(1+\dfrac{x-a}{b+a}\right)}^{r}[/latex] in the binomial expansion to find the Taylor series of each function with the given center.

3. [latex]\sqrt{{x}^{2}+2}[/latex] at [latex]a=0[/latex]
4. [latex]\sqrt{2x-{x}^{2}}[/latex] at [latex]a=1[/latex] (Hint: [latex]2x-{x}^{2}=1-{\left(x - 1\right)}^{2}[/latex])
5. [latex]\sqrt{x}[/latex] at [latex]a=4[/latex]
6. [latex]\sqrt{x}[/latex] at [latex]x=9[/latex]

In the following exercise, use the binomial theorem to estimate each number, computing enough terms to obtain an estimate accurate to an error of at most [latex]\dfrac{1}{1000}[/latex].

7. [latex]{\left(1001\right)}^{\dfrac{1}{3}}[/latex] using [latex]{\left(1000+x\right)}^{\dfrac{1}{3}}[/latex]

In the following exercises (8-10), use the binomial approximation [latex]\sqrt{1-x}\approx 1-\dfrac{x}{2}-\dfrac{{x}^{2}}{8}-\dfrac{{x}^{3}}{16}-\dfrac{5{x}^{4}}{128}-\dfrac{7{x}^{5}}{256}[/latex] for [latex]|x|<1[/latex] to approximate each number. Compare this value to the value given by a scientific calculator.

8. [latex]\sqrt{5}=5\times \dfrac{1}{\sqrt{5}}[/latex] using [latex]x=\dfrac{4}{5}[/latex] in [latex]{\left(1-x\right)}^{\dfrac{1}{2}}[/latex]
9. [latex]\sqrt{6}[/latex] using [latex]x=\dfrac{5}{6}[/latex] in [latex]{\left(1-x\right)}^{\dfrac{1}{2}}[/latex]
10. Recall that the graph of [latex]\sqrt{1-{x}^{2}}[/latex] is an upper semicircle of radius [latex]1[/latex]. Integrate the binomial approximation of [latex]\sqrt{1-{x}^{2}}[/latex] up to order [latex]8[/latex] from [latex]x=-1[/latex] to [latex]x=1[/latex] to estimate [latex]\dfrac{\pi }{2}[/latex].

In the following exercises (11-14), use the expansion [latex]{\left(1+x\right)}^{\dfrac{1}{3}}=1+\dfrac{1}{3}x-\dfrac{1}{9}{x}^{2}+\dfrac{5}{81}{x}^{3}-\dfrac{10}{243}{x}^{4}+\cdots[/latex] to write the first five terms (not necessarily a quartic polynomial) of each expression.

11. [latex]{\left(1+4x\right)}^{\dfrac{4}{3}};a=0[/latex]
12. [latex]{\left({x}^{2}+6x+10\right)}^{\dfrac{1}{3}};a=-3[/latex]
13. Use the approximation [latex]{\left(1-x\right)}^{\dfrac{2}{3}}=1-\dfrac{2x}{3}-\dfrac{{x}^{2}}{9}-\dfrac{4{x}^{3}}{81}-\dfrac{7{x}^{4}}{243}-\dfrac{14{x}^{5}}{729}+\cdots[/latex] for [latex]|x|<1[/latex] to approximate [latex]{2}^{\dfrac{1}{3}}={2.2}^{\dfrac{-2}{3}}[/latex].
14. Find the [latex]99[/latex] th derivative of [latex]f\left(x\right)={\left(1+{x}^{4}\right)}^{25}[/latex].

In the following exercises (15-18), find the Maclaurin series of each function.

15. [latex]f\left(x\right)={2}^{x}[/latex]
16. [latex]f\left(x\right)=\dfrac{\sin\left(\sqrt{x}\right)}{\sqrt{x}},\left(x>0\right)[/latex],
17. [latex]f\left(x\right)={e}^{{x}^{3}}[/latex]
18. [latex]f\left(x\right)={\sin}^{2}x[/latex] using the identity [latex]{\sin}^{2}x=\dfrac{1}{2}-\dfrac{1}{2}\cos\left(2x\right)[/latex]

In the following exercises (19-22), find the Maclaurin series of [latex]F\left(x\right)={\displaystyle\int }_{0}^{x}f\left(t\right)dt[/latex] by integrating the Maclaurin series of [latex]f[/latex] term by term. If [latex]f[/latex] is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero.

19. [latex]F\left(x\right)={\tan}^{-1}x;f\left(t\right)=\dfrac{1}{1+{t}^{2}}=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}{t}^{2n}[/latex]
20. [latex]F\left(x\right)={\sin}^{-1}x;f\left(t\right)=\dfrac{1}{\sqrt{1-{t}^{2}}}=\displaystyle\sum _{k=0}^{\infty }\left(\begin{array}{c}\dfrac{1}{2}\hfill \ k\hfill \end{array}\right)\dfrac{{t}^{2k}}{k\text{!}}[/latex]
21. [latex]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}\dfrac{{x}^{n}}{\left(2n\right)\text{!}}[/latex]
22. [latex]F\left(x\right)={\displaystyle\int }_{0}^{x}\dfrac{\text{ln}\left(1+t\right)}{t}dt;f\left(t\right)=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\dfrac{{t}^{n}}{n+1}[/latex]

In the following exercises (23-26), compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of [latex]f[/latex].

23. [latex]f\left(x\right)=\tan{x}[/latex]
24. [latex]f\left(x\right)={e}^{x}\cos{x}[/latex]
25. [latex]f\left(x\right)={\sec}^{2}x[/latex]
26. [latex]f\left(x\right)=\dfrac{\tan\sqrt{x}}{\sqrt{x}}[/latex] (see expansion for [latex]\tan{x}[/latex])

In the following exercises (27-28), find the radius of convergence of the Maclaurin series of each function.

27. [latex]\dfrac{1}{1+{x}^{2}}[/latex]
28. [latex]\text{ln}\left(1+{x}^{2}\right)[/latex]