Question

does the sequence a sub n = (2^n)/(5^n) converge or diverge? If it converges what does it converge to?

Answer 1

Its a geometric series:
\[\sum_{n=0}^{\infty}\frac{2^n}{5^n}=\sum_{n=0}^{\infty}(\frac{2}{5})^n\]
So since 2/5<1 the series converges to:
\[\frac{1}{1-2/5}=\frac{5}{3}\]
However, if your sum is:
\[\sum_{n=1}^{\infty}(\frac{2}{5})^n \rightarrow \frac{2}{5}\sum_{n=1}^{\infty}(\frac{2}{5})^{n-1}=\frac{2}{5}\frac{5}{3}=\frac{2}{3}\]
Notice the only difference is the bottom limit of the sum. Because a geometric series has to start with a 1 so since n starts at 1 for the bottom sum then you have to factor out a 2/5 to make the sum start at 1 (i.e., (2/5)^n-1 => (2/5)^(1-1) => (2/5)^0 =>1)