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]
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ROC: [latex]1[/latex]; IOC: [latex]\left(0,2\right)[/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]
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ROC: [latex]12[/latex]; IOC: [latex]\left(-16,8\right)[/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]
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[latex]\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}}{{3}^{n+1}}{x}^{n}[/latex]; ROC: [latex]3[/latex]; IOC: [latex]\left(-3,3\right)[/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]
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integration: [latex]\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}}{2n+1}{\left(2x\right)}^{2n+1}[/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]
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[latex]{p}_{4}\left(x\right)={\left(x+3\right)}^{3}-11{\left(x+3\right)}^{2}+39\left(x+3\right)-41[/latex]; exact
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]
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[latex]\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}{\left(3x\right)}^{2n}}{2n\text{!}}[/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]
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[latex]\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}}{\left(2n\right)\text{!}}{\left(x-\frac{\pi }{2}\right)}^{2n}[/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]
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[latex]\displaystyle\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}}{n\text{!}}{x}^{2n}[/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]
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[latex]F\left(x\right)=\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}}{\left(2n+1\right)\left(2n+1\right)\text{!}}{x}^{2n+1}[/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]
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Answers may vary.
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]
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[latex]2.5\text{%}[/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].
