# How do you use the direct Comparison test on the infinite series sum_(n=1)^oo5/(2n^2+4n+3) ?

Oct 2, 2014

By making the denominator smaller,

$\frac{5}{2 {n}^{2} + 4 n + 2} \le \frac{5}{n} ^ 2$.

Since

${\sum}_{n = 1}^{\infty} \frac{5}{n} ^ 2 = 5 {\sum}_{n = 1}^{\infty} \frac{1}{n} ^ 2$

is a convergent p-series with $p = 2 > 1$,

${\sum}_{n = 1}^{\infty} \frac{5}{2 {n}^{2} + 4 n + 2}$

converges by Comparison Test.

I hope that this was helpful.