Chapter 7: Q 69. (page 615)

Find the values of x for which the series $\sum_{K = 0}^{\infty} {(\sin x)}^{k}$ converges.

Short Answer

Expert verified

The series $\sum_{K = 0}^{\infty} {(\sin x)}^{k}$ converges for all real numbers except for $x \in \{(2 n + 1) \frac{π}{2} | n \in ℕ\}$ .

Step by step solution

Step 1. Given information.

Given a series $\sum_{K = 0}^{\infty} {(\sin x)}^{k}$ .

Step 2. Find all values of x for which the series converges.

A geometric series is of the form $\sum_{k = 0}^{\infty} c r^{k}$ for some constants c and r.

Supposer is a non-zero real number, then $\sum_{k = 0}^{\infty} c r^{k}$ converges to $\frac{c}{1 - r}$ if and only if $|r| < 1$ .

Here, the series localid="1648832395698" $\sum_{K = 0}^{\infty} {(\sin x)}^{k}$ has $r = \sin x$ .

For the series to converge, $|\sin x| < 1$ .

Note that $- 1 \leq \sin x \leq 1$ .

So, $|\sin x| = 1$ if and only if localid="1648832424318" $x = (2 n + 1) \frac{π}{2}$ for all natural number n.

It follows that $\sum_{K = 0}^{\infty} {(\sin x)}^{k}$ converges for all real number except for $x = (2 n + 1) \frac{π}{2}$ where localid="1648832538629" $n \in ℕ$ .

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Find the values of x for which the series $\sum_{K = 0}^{\infty} {(\sin x)}^{k}$ converges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find all values of x for which the series converges.

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