mattgondek4

2020-12-16

Use the Limit Comparison Test to determine the convergence or divergence of the series.
$\sum _{n=1}^{\mathrm{\infty }}\frac{5}{{4}^{n}+1}$

Lacey-May Snyder

Given series, $\sum _{n=1}^{\mathrm{\infty }}\frac{5}{{4}^{n}+1}$
we have to determine the convergence or divergence of the given series.
Consider ${a}_{n}=\frac{5}{{4}^{n}+1}$
Since ${4}^{n}+1>{4}^{n}$
$⇒\frac{1}{{4}^{n}+1}<\frac{1}{{4}^{n}}$
$⇒\frac{5}{{4}^{n}+1}<\frac{5}{{4}^{n}}$
so, $0<{a}_{n}<\frac{5}{{4}^{n}}$
Thus, we can compare the given series $\sum _{n=1}^{\mathrm{\infty }}\frac{5}{{4}^{n}+1}$ with the geometric series
$\sum _{n=1}^{\mathrm{\infty }}\frac{5}{{4}^{n}}=5\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{4}^{n}}$
this geometric series converges since $|\frac{1}{4}|<1$
so the comparison test tells us that $\sum _{n=1}^{\mathrm{\infty }}\frac{5}{{4}^{n}+1}$ also converges.

Jeffrey Jordon