Section 11.9

Key Problems: 1-4, 10, 15, 16, 21, 26, 27, 36, 37

Problem 1

If the radius of convergence of the power series ${\sum }_{n=0}^{\infty }{c}_n{x}^n$ is $10$, what is the radius of convergence of the series ${\sum }_{n=1}^{\infty }n{c}_n{x}^{n-1}$? Why?

Problem 2

Suppose you know the series ${\sum }_{n=0}^{\infty }{b}_n{x}^n$ converges for $|x|<2$. What can you say about the following series? Why?

$$\sum _{n=0}^{\infty }\frac{b_n}{n+1}{x}^{n+1}$$

Problem 3

Find a power series representation for the function and determine the interval of convergence.

$$f(x)=\frac{1}{1+x}$$

Problem 4

$$f(x)=\frac{x}{1+x}$$

Problem 10

$$f(x)=\frac{x}{2x^2+1}$$

Problem 15

(a) Use differentiation to find a power series representation for

$$f(x)=\frac{1}{(1+x{)}^2}$$

What is the radius of convergence?

(b) Use part (a) to find a power series for

$$f(x)=\frac{1}{(1+x{)}^3}$$

(c) Use part (b) to find a power series for

$$f(x)=\frac{x^2}{(1+x{)}^3}$$

Problem 16

(a) Use Equation 1¹ to find a power series representation for $f(x)=\ln (1-x)$. What is the radius of convergence?

(b) Use part (a) to find a power series for $f(x)=x\ln (1-x)$.

(c) By putting $x=\frac{1}{2}$ in your result from part (a), express $\ln 2$ as the sum of an infinite series.

Problem 21

Find a power series representation for the function and determine the radius of convergence.

$$f(x)=\ln (5-x)$$

Problem 26

Find a power series representation for $f$, and graph $f$ and several partial sums $s_n(x)$ on the same screen. What happens as $n$ increases?

$$f(x)={\tan }^{-1}(2x)$$

Problem 27

Evaluate the indefinite integral as a power series. What is the radius of convergence?

$$\int \frac{t}{1-t^8}dt$$

Problem 36

Use the result of Example 5 to compute $\ln 1.1$ correct to four decimal places.

Problem 37

(a) Show that the function

$$f(x)=\sum _{n=1}^{\infty }\frac{x^n}{n!}$$

is a solution of the differential equation

$$f^{\prime }(x)=f(x)$$

(b) Show that $f(x)=e^x$

Footnotes

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