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Mathematics LibreTexts

10.R: Chapter 10 Review Exercises

  • Gilbert Strang & Edwin “Jed” Herman
  • OpenStax

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True or False? In exercises 1 - 4, justify your answer with a proof or a counterexample.

1) If the radius of convergence for a power series \displaystyle \sum_{n=0}^∞a_nx^n is 5, then the radius of convergence for the series \displaystyle \sum_{n=1}^∞na_nx^{n−1} is also 5.

Answer
True

2) Power series can be used to show that the derivative of e^x is e^x. (Hint: Recall that \displaystyle e^x=\sum_{n=0}^∞\frac{1}{n!}x^n.)

3) For small values of x, \sin x ≈ x.

Answer
True

4) The radius of convergence for the Maclaurin series of f(x)=3^x is 3.

In exercises 5 - 8, find the radius of convergence and the interval of convergence for the given series.

5) \displaystyle \sum_{n=0}^∞n^2(x−1)^n

Answer
ROC: 1; IOC: (0,2)

6) \displaystyle \sum_{n=0}^∞\frac{x^n}{n^n}

7) \displaystyle \sum_{n=0}^∞\frac{3nx^n}{12^n}

Answer
ROC: 12; IOC: (−16,8)

8) \displaystyle \sum_{n=0}^∞\frac{2^n}{e^n}(x−e)^n

In exercises 9 - 10, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.

9) f(x)=\dfrac{x^2}{x+3}

Answer
\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{3^{n+1}}x^n; ROC: 3; IOC: (−3,3)

10) f(x)=\dfrac{8x+2}{2x^2−3x+1}

In exercises 11 - 12, find the power series for the given function using term-by-term differentiation or integration.

11) f(x)=\tan^{−1}(2x)

Answer
integration: \displaystyle \sum_{n=0}^∞\frac{(−1)^n}{2n+1}(2x)^{2n+1}

12) f(x)=\dfrac{x}{(2+x^2)^2}

In exercises 13 - 14, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?

13) f(x)=x^3−2x^2+4, \quad a=−3

Answer
p_4(x)=(x+3)^3−11(x+3)^2+39(x+3)−41; exact

14) f(x)=e^{1/(4x)}, \quad a=4

In exercises 15 - 16, find the Maclaurin series for the given function.

15) f(x)=\cos(3x)

Answer
\displaystyle \sum_{n=0}^∞\frac{(−1)^n(3x)^{2n}}{2n!}

16) f(x)=\ln(x+1)

In exercises 17 - 18, find the Taylor series at the given value.

17) f(x)=\sin x, \quad a=\frac{π}{2}

Answer
\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{(2n)!}\left(x−\frac{π}{2}\right)^{2n}

18) f(x)=\dfrac{3}{x},\quad a=1

In exercises 19 - 20, find the Maclaurin series for the given function.

19) f(x)=e^{−x^2}−1

Answer
\displaystyle \sum_{n=1}^∞\frac{(−1)^n}{n!}x^{2n}

20) f(x)=\cos x−x\sin x

In exercises 21 - 23, find the Maclaurin series for \displaystyle F(x)=∫^x_0f(t)\,dt by integrating the Maclaurin series of f(x) term by term.

21) f(x)=\dfrac{\sin x}{x}

Answer
\displaystyle F(x)=\sum_{n=0}^∞\frac{(−1)^n}{(2n+1)(2n+1)!}x^{2n+1}

22) f(x)=1−e^x

23) Use power series to prove Euler’s formula: e^{ix}=cosx+isinx

Answer
Answers may vary.

Exercises 24 - 26 consider problems of annuity payments.

24) For annuities with a present value of $1 million, calculate the annual payouts given over 25 years assuming interest rates of 1\%,5\%, and 10\%.

25) A lottery winner has an annuity that has a present value of $10 million. What interest rate would they need to live on perpetual annual payments of $250,000?

Answer
2.5\%

26) Calculate the necessary present value of an annuity in order to support annual payouts of $15,000 given over 25 years assuming interest rates of 1\%,5\%,and 10\%.


This page titled 10.R: Chapter 10 Review Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

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