## 4.1. Series and Convergence

### Examples 4.1.3(a):

The infinite series

The *= 1/2 + 1/4 + 1/8 + 1/16 + ...*converges to 1 (this series is a special case of the geometric series).*n*-th partial sum for this series is defined as

S_{n}= 1/2 + 1/2^{2}+ 1/2^{3}+ ... + 1/2^{n}

If we divide the above expression by 2 and then subtract it from the original one we get:

Hence, solving this forS_{n}- 1/2 S_{n}= 1/2 - 1/2^{n+1}

*S*we obtain

_{n}This is now a sequence, and we can take the limit asS_{n}= 2 (1/2 - 1/2^{n+1})

*n*goes to infinity. By our result on the power sequence, the term

*1/2*goes to zero, so that

^{n+1}That proves, by definition, that the infinite series converges to the number 1.lim S_{n}= 1