# Series and Infinite Series

Is the series sum from n=1 to infinity of (-1)^n convergent or divergent? Justify your answer. Can you now resolve the difficulty of the following:

Divergent series:

S=1+1/2+1/4+1/8+1/16+...

You have probably seen the following trick to sum this series: if we call the above sum S, then if we multiply by 2, we obtain: 2S=2+1+1/2+1/4+...=2+S

Hence S=2, so the series sumes to 2. However, if you apply the same trick to the series S=1+2+4+8+16+... one gets nonsensical results:

2s=2+4+8+16+...=S-1 => S=-1

So the same reasoning that shows that 1+1/2+1/4+...=2 also gives that 1+2+4+8+...=-1. Why is it that we trust the first equation but not the second? A similar example arises with the series:

S=1-1+1-1+1-1+...;

We can write S=1-(1-1+1-1+...)=1-S and that S=1/2

Instead, we can write S= (1-1)+(1-1)+(1-1)+...=0+0+... and hence S=0

Or, we can write S=1+(-1+1)+(-1+1)+...=1+0+0+...and S=1. Which one is correct?

https://brainmass.com/math/algebra/series-infinite-series-471165

#### Solution Summary

Series and infinite series are examined.