Prove the following statements about the geometric sum.

Question

Prove the following two statements:

1. $\Sigma^n_{i=0}q^i=\frac{q^{n+1}-1}{q-1}, n \in \mathbb{N}, q\neq1$
2. For a number $q$ with $|q|<1$, $\Sigma^ ∞_{i=0}q^i = \frac{1}{1-q}$ is true.

The first part is easy to prove. I simply used induction with a base case of $n=0$. However, I'm having trouble with the second part. How do I even go about proving it?

Answer 1

Since $\lim_{n \to \infty} q^n = 0$ when $|q| < 1 $, then we observe that

$$ \sum_{i \geq 0 } q^i = \lim_{i \to \infty} \sum_{k=0}^i q^k = \lim_{i \to \infty} \dfrac{ q^{i+1} - 1 }{q-1} = \dfrac{-1}{q-1} = \boxed{ \dfrac{1}{1-q} } $$