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Section 5.7

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1. (a) The interval of convergence is (-1, 1) .
The series converges at x = -1 and diverges at x = 1 .
The series converges absolutely on (-1, 1) , converges conditionally at x = -1 , and diverges for x < -1 and x \ge 1 .
\displaystyle{\sum_{n = 1}^\infty \frac{x^n}{n} = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} + \cdots}

(c) The interval of convergence is (-1, 1) .
The series diverges at both x = -1 and x = 1 .
The series converges absolutely on (-1, 1) and diverges for x \le -1 and x \ge 1 .
\displaystyle{\sum_{n = 1}^\infty nx^n = x + 2x^2 + 3x^3 + 4x^4 + 5x^5 + \cdots}

(e) The interval of convergence is (1, 3) .
The series converges at x = 1 and diverges at x = 3 .
The series converges absolutely on (1, 3) , converges conditionally at x = 1 , and diverges for x < 1 and x \ge 3 .
\displaystyle{\sum_{n = 0}^\infty \frac{(x-2)^{n+1}}{n+1} = (x - 2) + \frac{(x-2)^2}{2} + \frac{(x-2)^3}{3} + \frac{(x-2)^4}{4}}
\displaystyle{+ \frac{(x-2)^5}{5} + \cdots}

(g) The interval of convergence is (0, 2) .
The series converges at both x = 0 and x = 2 .
The series converges absolutely on [0, 2] and diverges for x < 0 and x > 2 .
\displaystyle{\sum_{n = 1}^\infty \frac{(x-1)^n}{n^2} = (x-1) + \frac{(x-1)^2}{4} + \frac{(x-1)^3}{9} + \frac{(x-1)^4}{16}}
\displaystyle{+ \frac{(x-1)^5}{25} + \cdots}

2. (a) The interval of convergence is (-\infty, \infty) .
The series converges absolutely on (-\infty, \infty) .
\displaystyle{\sum_{n = 0}^\infty \frac{(x-3)^n}{n!} = 1 + (x - 3) + \frac{(x-3)^2}{2} + \frac{(x-3)^3}{6} + \frac{(x-3)^4}{24} + \cdots}

(c) The interval of convergence is (-\sqrt{2}, \sqrt{2}) .
The series diverges at both x = -\sqrt{2} and x = \sqrt{2} .
The series converges absolutely on (-\sqrt{2}, \sqrt{2}) and diverges for x \le -\sqrt{2} and x \ge \sqrt{2} .
\displaystyle{\sum_{n = 0}^\infty \frac{x^{2n}}{2^n} = 1 + \frac{x^2}{2} + \frac{x^4}{4} + \frac{x^6}{8} + \frac{x^8}{16}+ \cdots}

(e) The interval of convergence is (-1, 1) .
The series diverges at both x = -1 and x = 1 .
The series converges absolutely on (-1, 1) and diverges for x \le -1 and x \ge 1 .
\displaystyle{\sum_{n = 1}^\infty \frac{x^{2n+1}}{n} = x^3 + \frac{x^5}{2} + \frac{x^7}{3} + \frac{x^9}{4} + \frac{x^{11}}{5} + \cdots}

(g) The interval of convergence is \displaystyle{\left(-\frac{1}{3}, \frac{1}{3}\right)}.
The series diverges at both x = -\dfrac{1}{3} and x = \dfrac{1}{3} .
The series converges absolutely on \displaystyle{\left(-\frac{1}{3}, \frac{1}{3}\right)} and diverges for x \le -\dfrac{1}{3} and x \ge \dfrac{1}{3} .
\displaystyle{\sum_{n = 1}^\infty 3^nx^n = 3x + 9x^2 + 27x^3 + 81x^4 + 243x^5 + \cdots}

3. (a) For -1 < x < 1 ,

\displaystyle{\frac{1}{1 + x} = \sum_{n=0}^\infty (-1)^nx^n = 1 - x + x^2 - x^3 + x^4 - \cdots.}

(b) f^{(35)}(0) = -35!

(c) \displaystyle{\int_0^x \frac{1}{1 + t} \ dt = \sum_{n=0}^\infty \frac{(-1)^n x^{n+1}}{n + 1} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \cdots}
The series converges absolutely on (-1, 1) , converges conditionally at x = 1 , and diverges for x \le -1 and x > 1 .

(d) \displaystyle{\int_0^1 \frac{1}{1 + t} \ dt = \sum_{n=0}^\infty \frac{(-1)^n}{n + 1} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots}
\displaystyle{\int_0^1 \frac{1}{1 + t} \ dt \approx \sum_{n=0}^{998} \frac{(-1)^n}{n + 1} = 0.6936}, rounded to 4 decimal places

5. \displaystyle{\cos(1) = \sum_{n=0}^\infty \frac{(-1)^{2n}}{(2n)!} = 1 - \frac{1}{2} + \frac{1}{4!} - \frac{1}{6!} + \frac{1}{8!} - \cdots}
\displaystyle{\cos(1) \approx 1 - \frac{1}{2} + \frac{1}{24} - \frac{1}{720} + \frac{1}{40,320} = 0.5403026}, rounded to 7 decimal places

7. \displaystyle{\sum_{n=1}^\infty \frac{n}{2^{n-1}} = \frac{1}{\left(1 - \frac{1}{2}\right)^2} = 4}

8. (a) \displaystyle{A = \sum_{n=1}^\infty \frac{n}{2^n} = \frac{1}{2}\sum_{n=1}^\infty \frac{n}{2^{n-1}} = 2}

(b) \displaystyle{A = q\sum_{n=1}^\infty np^{n-1} = \frac{q}{(1 - p)^2} = \frac{1}{q}}

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