Question

Evaluate $\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{1}{2^r}$ where $[.]$ denotes the greatest integer function.

Answer 1

${\sum }_{r=1}^n\frac{1}{2^r}$

$r=\frac{1}{2},a={\left(\frac{1}{2}\right)}^r$

$a_1\frac{1-r^r}{1-r}$

$=\frac{1}{2}\cdot \frac{1-{\left(\frac{1}{2}\right)}^r}{1-\frac{1}{2}}$

$=1-{\left(\frac{1}{2}\right)}^n$

${lim}_{n\to \infty }{\sum }_{r=1}^n\frac{1}{2^r}$

$={lim}_{n\to \infty }(1-{\left(\frac{1}{2}\right)}^n)$

$={lim}_{n\to \infty }(1\cdot (1-\frac{{\left(\frac{1}{2}\right)}^n}{1}))$

$={lim}_{n\to \infty }\left(1\right)\cdot {lim}_{n\to \infty }(1-\frac{{\left(\frac{1}{2}\right)}^n}{1})$

$=1$