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:
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:
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?
Series and infinite series are examined.