Question

Fun problem of the day,

Find the sum of the series \[ \frac 1 2 + \frac 3 4 + \frac 5 8 +\frac 7 {16} \cdots \]

Enjoy!

Answer 1

\[\sum_{n=1}^{\infty}\frac{2n-1}{2^n}=\sum_{n=1}^{\infty}\frac{n}{2^{n-1}}-\sum_{n=1}^{\infty}\frac{1}{2^{n}}\]

Answer 2

For the second part,\[\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}\]Note the 0 on the bottom, take away the term for n=0 for your example.

Answer 3

Oh, sorry, you're not asking!