Limit Comparison Test of the series of sin(1/n)

Printable View

December 20th 2012, 01:39 PM
MSUMathStdnt

Limit Comparison Test of the series of sin(1/n)

What series do I compare $\sum_{n=1}^\infty \sin \frac 1n$ to when using the Limit Comparison test.
December 20th 2012, 01:44 PM
Plato

Re: Limit Comparison Test of the series of sin(1/n)

Quote:

Originally Posted by MSUMathStdnt

What series do I compare $\sum_{n=1}^\infty \sin \frac 1n$ to when using the Limit Comparison test.

$\sum_{n=1}^\infty \frac{ 1}{n}$
December 20th 2012, 02:27 PM
SworD

Re: Limit Comparison Test of the series of sin(1/n)

But that won't be sufficient, because for large n, $\sin{\frac{1}{n}} < \frac{1}{n}$

$\sum_{n=1}^\infty \frac{ 1}{n}$ diverges, but the series in the original post consists of smaller terms, so you can't use that as a direct comparison. You can however, use the fact that

$\sin{\frac{1}{n}} > \frac{1}{2n}$

and the fact that the below diverges:

$\sum_{n=1}^\infty \frac{ 1}{2n}$

In fact the 2 in the above series can be ANY number greater than 1.
December 20th 2012, 02:46 PM
Plato

Re: Limit Comparison Test of the series of sin(1/n)

Quote:

Originally Posted by SworD

But that won't be sufficient, because for large n, $\sin{\frac{1}{n}} < \frac{1}{n}$

@SworD, you did not read the question carefully.

The question is about limit comparison not basic comparison.
Some authors like Gillman call it ratio comparison.

$\lim _{n \to \infty } \frac{{\sin \left( {\frac{1}{n}} \right)}}{{\frac{1}{n}}} = 1$
December 20th 2012, 08:13 PM
hollywood

Re: Limit Comparison Test of the series of sin(1/n)

Yes, the (ordinary) comparison test is if $0 \le a_n \le b_n$ where $\sum_{n=1}^\infty{b_n}$ is a convergent series, then $\sum_{n=1}^\infty{a_n}$ converges. Or if $0 \le b_n \le a_n$ where $\sum_{n=1}^\infty{b_n}$ is a divergent series, then $\sum_{n=1}^\infty{a_n}$ diverges.

For the limit comparison test, if $\lim_{n \rightarrow \infty}\frac{a_n}{b_n}$ is finite and nonzero, then $\sum_{n=1}^\infty{a_n}$ converges if and only if $\sum_{n=1}^\infty{b_n}$ converges.

So you need to compare $\sin(1/n)$ to 1/n, since $\lim_{n \rightarrow \infty}\frac{\sin(1/n)}{1/n}=1$ is finite and nonzero. Of course, 1/2n or 35/87n would also work - the limits would still be finite and nonzero.

- Hollywood