Why are we interested in Infinite Series?

Example 1:

log 2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - + ...

Example 2:

e = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + ... + 1/(n!) + ...