Question

(5CQ) Use the comparison test to determine whether the series 25/3+125/9+625/27+... is convergent or divergent.

Answer 1

The terms in the sum are given by the sequence

$\left\{\dfrac{5^{n+1}}{3^n}\right\}_{n\ge1}$

We have

$\dfrac{5^{n+1}}{3^n}>\dfrac{5^n}{3^n}=\left(\dfrac53\right)^n>1$

for all $n$, and the series $1+1+1+\cdots$ clearly diverges, so $\dfrac{25}3+\dfrac{125}9+\cdots$ must also diverge.