arithmetic/geometric series

a=sumfor(n, 1, 1000000,  1 + (2*n+1) / 5^n)==It converges to 1 + 7/8 

\(\displaystyle S = 1 + \frac{3}{5}+\frac{5}{5^{2}}+\frac{7}{5^{3}}+ \dots \displaystyle = 1+\frac{1+2}{5}+\frac{3+2}{5^{2}}+\frac{5+2}{5^{3}}+\dots \\ \displaystyle = 1 + \frac{1}{5}+\frac{3}{5^{2}}+\frac{5}{5^{3}}+\dots + \frac{2}{5}+\frac{2}{5^{2}}+\frac{2}{5^{3}}+\dots \\ \displaystyle = 1+\frac{1}{5}\left \{1+\frac{3}{5}+\frac{5}{5^{2}}+ \dots \right \} +\frac{2}{5}\left \{1+\frac{1}{5}+\frac{1}{5^{2}}+\frac{1}{5^{3}}+ \dots \right \} \\ \displaystyle =1+\frac{1}{5}S+\frac{2}{5}.\frac{1}{1-(1/5)} \\ \displaystyle \frac{4}{5}S=\frac{3}{2}, \quad S = \frac{15}{8}.\)