# examples using comparison test without limit

Do the following series converge?

 $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{2}+n+1}$ (1)
 $\displaystyle\sum_{n=1}^{\infty}\frac{n^{3}+n+1}{n^{4}+n+1}$ (2)

The general of (1) may be estimated upwards:

 $0<\frac{1}{n^{2}+n+1}<\frac{1}{n^{2}+0+0}=\frac{1}{n^{2}}$

Because  $\sum_{n=1}^{\infty}\frac{1}{n^{2}}$ (an over-harmonic series) converges, then also (1) converges.

The general of (2) may be estimated downwards:

 $\frac{n^{3}+n+1}{n^{4}+n+1}>\frac{n^{3}+0+0}{n^{4}+n^{4}+n^{4}}=\frac{1}{3}% \cdot\frac{1}{n}>0$

Because $\sum_{n=1}^{\infty}\frac{1}{3}\frac{1}{n}$ (the harmonic series divided by 3) diverges, then also (2) diverges.

Title examples using comparison test without limit ExamplesUsingComparisonTestWithoutLimit 2013-03-22 15:08:55 2013-03-22 15:08:55 pahio (2872) pahio (2872) 9 pahio (2872) Example msc 40-00 PTest over-harmonic series