2.6: Chapter 2 Review Exercises

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 \(10\%.\)