11.9E: Exercises for Section 11.9

In the following exercises, use the fact that \( \displaystyle \frac{1}{1-u} = \sum_{n=0}^∞ u^n \ \) to

find a power series for each function with the given center \( a \). State the radius of convergence.

1) \(\displaystyle f(x) = \frac{1}{x}; \; a=1 \)

Answer

Hint: write \( \displaystyle f(x) = \frac{1}{x} = \frac{1}{1-(1-x)} = \sum_{n=0}^∞ (1-x)^n; \; R=1 \)

2) \(\displaystyle f(x) = \frac{1}{1-x^2}; \;a=0 \)

Answer

\( \displaystyle f(x) = \sum_{n=0}^∞ x^{2n}; \; R=1 \)

3) \(\displaystyle f(x) = \frac{x}{1-x^2}; \;a=0 \)

Answer

\( \displaystyle f(x) = \sum_{n=0}^∞ x^{2n+1}; \; R=1 \)

4) \(\displaystyle f(x) = \frac{1}{1+x^2}; \;a=0 \)

Answer

\( \displaystyle f(x) = \sum_{n=0}^∞ (-1)^n x^{2n}; \; R=1 \)

5) \(\displaystyle f(x) = \frac{x^2}{1+x^2}; \;a=0 \)

Answer

\( \displaystyle f(x) = \sum_{n=0}^∞ (-1)^n x^{2n+2}; \; R=1 \)

6) \(\displaystyle f(x) = \frac{1}{2-x}; \;a=1 \)

Answer

\( \displaystyle f(x) = \sum_{n=0}^∞ (x-1)^{n}; \; R=1 \)

7) \(\displaystyle f(x) = \frac{1}{1-2x}; \;a=0 \)

Answer

\( \displaystyle f(x) = \sum_{n=0}^∞ 2^n x^{n}; \; R= \frac 1 2 \)

8) \(\displaystyle f(x) = \frac{1}{1-4x^2}; \;a=0 \)

Answer

\( \displaystyle f(x) = \sum_{n=0}^∞ 2^{2n} x^{2n}; \; R= \frac 1 2 \)

9) \(\displaystyle f(x) = \frac{x^2}{1-4x^2}; \;a=0 \)

Answer

\( \displaystyle f(x) = \sum_{n=0}^∞ 2^{2n} x^{2n+2}; \; R= \frac 1 2 \)

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