Show that the sum of the terms of a sequence converges

Question

The sequence is $u_n=(1-a)^n$ with $0< a < 1$. This sequence clearly converges to $0$. How can I demonstrate that the sum of its terms converges? Thank you.

Answer 1

Let $b:=1-a$. What can you say about $(1-b)*(1+b+b^2+\ldots +b^n)$? Hint: Multiply out.

Then observe that $b^n$ goes to $0$ if $n$ goes to infinity. Finally you get $\sum_{n=0}^{\infty} u_n = \sum_{n=0}^{\infty} b^n=?$