Question

Sum to $n$ terms of the following series:
$1+(1+\frac{1}{2})+(1+\frac{1}{2}+\frac{1}{4})+,...$

Answer 1

$t_n=(1+\frac{1}{2}+\frac{1}{4}+.....ntimes)$
$=\frac{1(1-{\left(\frac{1}{2}\right)}^n)}{(1-\frac{1}{2})}=2(1-\frac{1}{2n})$
$tn=\sum tn$
$=\sum 2(1-\frac{1}{2n})=\sum 2-\frac{2}{2n}$
$=2n-2(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{263}+....ntimes)$
$=2n-2\left(\frac{\frac{1}{2}(1-\frac{1}{2n})}{(1-\frac{1}{2})}\right)$
$=2n-n(1-\frac{1}{2n})$