Limit of expression with increasing exponent

Question

I got stuck trying to evaluate: $$\lim\limits_{n\to \infty}\left(\frac{\sqrt2}{\sqrt2+1}\right) + \left(\frac{\sqrt2}{\sqrt2+1}\right)^2 + \left(\frac{\sqrt2}{\sqrt2+1}\right)^3 + \dots + \left(\frac{\sqrt2}{\sqrt2+1}\right)^n $$

I'd be grateful if you gave me some tips.

Answer 1

Hint : $$\sum_0^N x^n= \frac{x^{N+1}-1}{x-1}$$