Question

Find the sum of the series $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.......\infty .$

Answer 1

$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.....\infty r=\frac{1}{2}a=1s_n=\frac{a(1-r^n)}{(1-r)}{s}_n=\frac{a(1-(\frac{1}{2}{)}^n)}{(1-\frac{1}{2})}=2(1-{\left(\frac{1}{2}\right)}^n)(2-2^{1-n})\approx 2$

As $2^{1-n}$ will tens to $0$ as $n\to \infty$