# Math Help - Integral Test/Comparison Test

1. ## Integral Test/Comparison Test

infinity over Sigma n=1 n^2/n^3 + 1 Use the integral test to see if the infinite series converges.

infinity over Sigma n=2 n^2+1/n^3.5-2 Use either the Comparison Test or the Limit Comparison Test to see if the infinite series converges.

Put $f(x)=\frac{{{x}^{3}}}{{{x}^{2}}+1},$ then $f$ is a positive, continuous and decreasing function for $x\ge2,$ hence, the integral test applies and the series will converge or diverge if $\int_{2}^{\infty }{\frac{{{x}^{3}}}{{{x}^{2}}+1}\,dx}$ does.
Since $\frac{{{x}^{3}}}{{{x}^{2}}+1}\ge \frac{{{x}^{3}}}{{{x}^{2}}+{{x}^{2}}}=\frac{1}{2}x ,$ then the integral diverges, so does the series. (Actually, one can bound immediately the general term of the series rather than applying the integral test.)
As for your second question, compare the series with $\sum\limits_{n=2}^{\infty }{\frac{1}{{{n}^{1,5}}}}$ which is a convergent $p-$series with $p=1,5>1.$
So, put ${{a}_{n}}=\frac{{{n}^{2}}+1}{{{n}^{3,5}}-2}$ and ${{b}_{n}}=\frac{1}{{{n}^{1,5}}},$ and observe that $\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{{{b}_{n}}}=1,$ so your series converges.