# Lab 1. Exercise 1
###### tags: `lab1`

$$
S = \sum_{n=1}^{\infty}(f_n - f_{n+1})
$$

Define partial sum $S_N = \sum_{n=1}^{N}(f_n - f_{n+1})$. Expanding sigma we obtain:

$$
S_N = f_1 - f_2 + f_2 - f_3 + \dots + f_{N} - f_{N+1} = f_1 - f_{N+1}
$$

Sum of the infinite series is defined as a limit of the partial sums: $S = \lim_{N \to \infty} S_N = \lim_{N \to \infty} (f_1 - f_{N+1}) = f_1 - lim_{N \to \infty} f_{N+1}$

**Note:** assuming that $\{f_n\}$ converges to some finite number $f$ we can re-write the last equality as following:

$S = 1 - f$.