Why does
\(\sum_{n=0}^{\infty}{2^{-n}}=2\)
but
\(\sum_{n=1}^{\infty}{n^{-1}}=\infty??????????????\)
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.
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 + ...
+776 
