# Limit Comparison Test vs Comparison Test

In order to test the convergence of $\sum_{k=1}^{\infty}\frac{\sin^2k}{k^2}$, it is rather easy to comapre $\frac{\sin^2k}{k^2}$ with $\frac{1}{k^2}$ and use the Comparison Test ($0\le \frac{\sin^2k}{k^2}\le \frac{1}{k^2}$). On the other hand, since $$\lim_{k \to \infty}\frac{\frac{\sin^2k}{k^2}}{\frac{1}{k^2}} =\lim_{k \to \infty}\sin^2k$$ does not exist and Limit Coparison test cannot be used if we consider $\frac{1}{k^2}$.

I wonder if it is possible to use Limit Comparison Test by using other $b_k$?

Edit: The Limit Comparison Test that I refer to is only consider if $\lim_{k=1}^{\infty}\frac{a_k}{b_k}$ is a positive number. See (http://mathworld.wolfram.com/LimitComparisonTest.html). May be I should rephrase my question as follow.

Does there exist $b_k$ such that $\displaystyle\lim_{k \to \infty}\frac{\left(\frac{\sin^2k}{k^2}\right)}{b_k}$ is a finite positive number and $\sum_{k=1}^{\infty}b_k$ is convergent?

• I do not know how useful can be this but suppose you consider at the same time $\sum_{k=1}^{\infty}\frac{\cos^2k}{k^2}$, the sum of the two sums is exactly $\frac{\pi ^2}{6}$. Just for your curiosity, the value of the sum you consider is exactly $\frac{1}{2} (\pi -1)$ Feb 15, 2014 at 4:34
• @ClaudeLeibovici Thanks for the info. Is there any reference or hint of getting the exact sum of the series?
– pipi
Feb 15, 2014 at 15:21
• @pipi.May be Wolfram Alpha or another CAS. Feb 15, 2014 at 15:23
• @ClaudeLeibovici Oh yes. I just tried Wolfram Alpha. I wonder how to calculate it in mathematical way...
– pipi
Feb 15, 2014 at 15:33
• Ummmmmmmh, I really don't know ! I shall work on that. Cheers. Feb 15, 2014 at 15:58

Sure, use $b_k=\frac{1}{k^{3/2}}$. The limit is of the ratio is $0$, and since $\frac{3}{2}\gt 1$, $\sum_1^\infty \frac{1}{k^{3/2}}$ converges.

• Nicolas Thanks for the answer. I rephrase my question.
– pipi
Feb 15, 2014 at 15:23

There's a different statement of the limit comparison test that would work:

Let $\{{a_n}\}$ and $\{{b_n}\}$ be sequences of nonnegative reals. Then the following hold (the first part is what you can use):

If $\limsup_{k \to \infty}\frac{a_k}{b_k}$ is finite, then convergence of $\{{b_n}\}$ implies convergence of $\{{a_n}\}$.

If $\liminf_{k \to \infty}\frac{a_k}{b_k} > 0$, then divergence of $\{{b_n}\}$ implies divergence of $\{{a_n}\}$.)