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# 10R: Chapter 10 Review Exercises

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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 $$5$$.

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 $$x,$$ $$\sin x ≈ x.$$

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

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

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

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

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

$$\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{3^{n+1}}x^n;$$ 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) $$f(x)=\tan^{−1}(2x)$$

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) $$f(x)=x^3−2x^2+4, \quad a=−3$$

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

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

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

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

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

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

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

17) $$f(x)=\sin x, \quad a=\frac{π}{2}$$

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

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

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

19) $$f(x)=e^{−x^2}−1$$

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

20) $$f(x)=\cos x−x\sin x$$

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

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

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

22) $$f(x)=1−e^x$$

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

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 $$250,000$$?

$$2.5\%$$
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 $$10\%.$$