# examples for limit comparison test

###### Example 1.

Does the following series converge?

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

The series is similar to $\sum_{n=1}^{\infty}1/n^{2}$ which converges (use $p$-test, for example). Next we compute the limit:

 $\lim_{n\to\infty}\frac{\frac{1}{n^{2}+n+1}}{\frac{1}{n^{2}}}=\lim_{n\to\infty}% \frac{n^{2}}{n^{2}+n+1}=1$

Therefore, since $1\neq 0$, by the Limit Comparison Test (with $a_{n}=1/(n^{2}+n+1)$ and $b_{n}=1/n^{2}$), the series converges.

###### Example 2.

Does the following series converge?

 $\sum_{n=1}^{\infty}\frac{n^{3}+n+1}{n^{4}+n+1}$

If we “forget” about the lower order terms of $n$:

 $\frac{n^{3}+n+1}{n^{4}+n+1}\sim\frac{n^{3}}{n^{4}}=\frac{1}{n}$

and $\sum_{n=1}^{\infty}1/n$ is the harmonic series which diverges (by the $p$-test). Thus, we take $b_{n}=1/n$ and compute:

 $\lim_{n\to\infty}\frac{\frac{n^{3}+n+1}{n^{4}+n+1}}{\frac{1}{n}}=\lim_{n\to% \infty}\frac{n(n^{3}+n+1)}{n^{4}+n+1}=\lim_{n\to\infty}\frac{n^{4}+n^{2}+n}{n^% {4}+n+1}=\lim_{n\to\infty}\frac{1+1/n^{2}+1/n^{3}}{1+1/n^{3}+1/n^{4}}=1$

Therefore the series diverges like the harmonic does.

Title examples for limit comparison test ExamplesForLimitComparisonTest 2013-03-22 15:08:48 2013-03-22 15:08:48 alozano (2414) alozano (2414) 4 alozano (2414) Example msc 40-00