Prove or disprove that the series $\sum_{n=1}^\infty \frac{1}{4n^2 +7}$ is convergent. [closed]
How can I prove or disprove that the series $\sum_{n=1}^\infty \frac{1}{4n^2 +7}$ is convergent?
Solution 1:
Notice the term of $n^2$ in the bottom. That should give us a hint that maybe we want to try comparing to something we already know.
We know that $$ \sum_{n=1}^\infty \frac{1}{n^2} $$
is a convergent sum. Since $\forall n\geq 0$, $4n^2+7 > n^2$ we know that $$ \frac{1}{4n^2+7} < \frac{1}{n^2}. $$ By extension we can say that $$ \sum_{n=1}^\infty \frac{1}{4n^2+7} < \sum_{n=1}^\infty \frac{1}{n^2}, $$ which means that by the direct comparison test our sum is convergent