Huh?

Question

0

867

2

+776

Why does

$\sum_{n=0}^{\infty}{2^{-n}}=2$

but

$\sum_{n=1}^{\infty}{n^{-1}}=\infty??????????????$

User101 May 15, 2016

Guest · Answer 1 · 2016-05-15T04:41:10

1) Because: Sum of: 1/1 + 1/2 + 1/4 + 1/8 + 1/16..........converges to 2!

2) Because: Sum of: 1/1 + 1/2 + 1/3 + 1/4 + 1/5............diverges to infinity. This is the Harmonic series.

Alan · Answer 2 · 2016-05-15T04:55:59

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

Write it as: S = 1 + (1/2)*(1 + 1/2 + 1/4 + 1/8 + 1/16 + ...)

or S = 1 + (1/2)*S

Subtract S/2 from both sides: S/2 = 1

Multiply both sides by 2: S = 2

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