Here is the question for context:
The number of claims occurring in a period has a Poisson distribution with mean $\lambda$. the insurer determines the conditional expectation of the number of claims in the period given that at least one claim has occurred, say $e(\lambda)$. Find $\lim_{\lambda\to 0} e(\lambda)$.
So the solution is provided, I'll point out the part I don't understand with an under-brace and question mark with my specific questions below:
Solution: $E[N|N\geq 1]=\sum_{i=1}^{\infty }n\cdot f(n|N\geq 1)=\underbrace{\sum_{i=1}^{\infty }n\cdot \frac{f(n)}{1-f(0)}=\sum_{i=0}^{\infty }n\cdot \frac{f(n)}{1-f(0)}}_{???}$
My specific question is: Why is it ok to change the summation index from 1 to 0 without changing anything else about the expression?
\begin{align} E[N|N\geq 1]&=\sum_{n=0}^{\infty }n\cdot f(n|N\geq 1)= \\ &=\sum_{n=1}^{\infty }n\cdot f(n|N\geq 1)= \\ &={\sum_{n=1}^{\infty }n\cdot \frac{f(n)}{1-f(0)}=0 \frac{f(0)}{1-f(0)} + \sum_{n=1}^{\infty }n\cdot \frac{f(n)}{1-f(0)}= \sum_{n=0}^{\infty }n\cdot \frac{f(n)}{1-f(0)}} \\ &= \frac{E[N]}{1-P(N=0)} \end{align}