Question

Find the following limit.
$\underset{n\to \infty }{lim}\frac{1+5+5^2+...+5^{n-1}}{1-(25{)}^n}.$

Answer 1

$\underset{n\to \infty }{lim}\frac{1+5+5^2+.........+5^{n-1}}{(1-5^n)(1+5^n)}=\underset{n\to \infty }{lim}\frac{(5^n-1)}{4(1-5^n)(1+5^n)}=\underset{n\to \infty }{lim}\frac{-1}{4(1+5^n)}=0$