# Thread: Convergence of series with trigonometric function :(

1. ## Convergence of series with trigonometric function :(

hello, this is the problem:

show that

infinite
Σ sin (1/n^2) is a positive, convergent series.
n=1

Hint: use the inequality sinx is less than or equal to x for x is greater than or equal to 0

2. ## p-series

Are you familiar with the convergence/divergence of the p-series $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^p}$?

--Kevin C.

3. yes, but can this be applied even though the exponent is within the trigonometric function?

4. Originally Posted by wyhwang7 yes, but can this be applied even though the exponent is within the trigonometric function?
Not exactly. First show that 1/n^2 is a converging series. Then use the hint the problem gave, that for n>0, $\displaystyle \sin(n) \le n$. Put another way, $\displaystyle \frac{1}{\sin^2(n)} \le \frac{1}{n^2}$ So we have established a bounding series that converges and this proves that the other series converges as well.

5. I made a typo. The sine should be around the whole expression, not just the denominator. The way I wrote it was wrong, sorry.

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