proof verification - prove that $1+x+x^2+...x^n=frac{1-x^{n+1}}{1-x} $

Prove that $1+x+x^2+...x^n=\frac{1-x^{n+1}}{1-x} $

and how to prove this implies that for $|x|<1$

$$1+x+x^2+........=\frac{1}{1-x}$$

for the first part we can prove by induction but to get second one by using first one

Answer

Hint:

(Both for the induction part and for the limit).

Rewrite the r.h.s. as $\;\dfrac1{1-x}-\dfrac{x^{n+1}}{1-x}$.