Module 6 Review Problems

True or False? In the following exercises, justify your answer with a proof or a counterexample.

1. If the radius of convergence for a power series [latex]\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}[/latex] is [latex]5[/latex], then the radius of convergence for the series [latex]\displaystyle\sum _{n=1}^{\infty }n{a}_{n}{x}^{n - 1}[/latex] is also [latex]5[/latex].

2. Power series can be used to show that the derivative of [latex]{e}^{x}\text{ is }{e}^{x}[/latex]. (Hint: Recall that [latex]{e}^{x}=\displaystyle\sum _{n=0}^{\infty }\frac{1}{n\text{!}}{x}^{n}.[/latex])

3. For small values of [latex]x,\sin{x}\approx x[/latex].

4. The radius of convergence for the Maclaurin series of [latex]f\left(x\right)={3}^{x}[/latex] is [latex]3[/latex].

In the following exercises, find the radius of convergence and the interval of convergence for the given series.

5. [latex]\displaystyle\sum _{n=0}^{\infty }{n}^{2}{\left(x - 1\right)}^{n}[/latex]

6. [latex]\displaystyle\sum _{n=0}^{\infty }\frac{{x}^{n}}{{n}^{n}}[/latex]

7. [latex]\displaystyle\sum _{n=0}^{\infty }\frac{3n{x}^{n}}{{12}^{n}}[/latex]

8. [latex]\displaystyle\sum _{n=0}^{\infty }\frac{{2}^{n}}{{e}^{n}}{\left(x-e\right)}^{n}[/latex]

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

9. [latex]f\left(x\right)=\frac{{x}^{2}}{x+3}[/latex]

10. [latex]f\left(x\right)=\frac{8x+2}{2{x}^{2}-3x+1}[/latex]

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

11. [latex]f\left(x\right)={\tan}^{-1}\left(2x\right)[/latex]

12. [latex]f\left(x\right)=\frac{x}{{\left(2+{x}^{2}\right)}^{2}}[/latex]

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

13. [latex]f\left(x\right)={x}^{3}-2{x}^{2}+4,a=-3[/latex]

14. [latex]f\left(x\right)={e}^{\frac{1}{\left(4x\right)}},a=4[/latex]

In the following exercises, find the Maclaurin series for the given function.

15. [latex]f\left(x\right)=\cos\left(3x\right)[/latex]

16. [latex]f\left(x\right)=\text{ln}\left(x+1\right)[/latex]

In the following exercises, find the Taylor series at the given value.

17. [latex]f\left(x\right)=\sin{x},a=\frac{\pi }{2}[/latex]

18. [latex]f\left(x\right)=\frac{3}{x},a=1[/latex]

In the following exercises, find the Maclaurin series for the given function.

19. [latex]f\left(x\right)={e}^{\text{-}{x}^{2}}-1[/latex]

20. [latex]f\left(x\right)=\cos{x}-x\sin{x}[/latex]

In the following exercises, find the Maclaurin series for [latex]F\left(x\right)={\displaystyle\int }_{0}^{x}f\left(t\right)dt[/latex] by integrating the Maclaurin series of [latex]f\left(x\right)[/latex] term by term.

21. [latex]f\left(x\right)=\frac{\sin{x}}{x}[/latex]

22. [latex]f\left(x\right)=1-{e}^{x}[/latex]

23. Use power series to prove Euler’s formula: [latex]{e}^{ix}=\cos{x}+i\sin{x}[/latex]

The following exercises consider problems of annuity payments.

24. For annuities with a present value of [latex]$1[/latex] million, calculate the annual payouts given over [latex]25[/latex] years assuming interest rates of [latex]1\text{%},5\text{%},\text{and }10\text{%}[/latex].

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

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