Answer
Converges
Work Step by Step
Here, we have the given series as $\dfrac{1}{ 5^{n}}$
Here, we can see that the function $F(n)=\dfrac{1}{ 5^{n}}$ shows a positive and continuous for $n \geq 1$.
Next, we have $F(n)=\dfrac{1}{ 5^{n}} \implies F'(n)=\dfrac{-n}{5^{n+1}}$. We can see that $F'(n) \lt 0 $ for $n \gt 1$. This implies that the given function is decreasing. Thus, we need to apply the integral test.
So, $\int_1^{\infty} \dfrac{dn}{5^n}=\lim\limits_{a \to \infty} \int_1^{a} \dfrac{dn}{5^n}$
or, $=\lim\limits_{a \to \infty} [-\dfrac{5^{-n}}{\ln 5}]_1^{\infty}$
or, $=\dfrac{1}{5 \ln (5)}$
Therefore, the series converges by the integral test.
