Question

# Find the sum of the series, if it converges: a) sum_(k = 0)^oo (1/3)^k b) sum_(k = 0)^oo (-1/2)^k

Series
Find the sum of the series, if it converges:
a) $$\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\left(\frac{{1}}{{3}}\right)}^{{k}}$$
b) $$\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\left(-\frac{{1}}{{2}}\right)}^{{k}}$$

2021-09-06

a) $$\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\left(\frac{{1}}{{3}}\right)}^{{k}}$$
A series of form $$\displaystyle{r}^{{k}}$$ converges for $$\displaystyle{\left|{r}\right|}{<}{1}$$
Here, $$\displaystyle{r}=\frac{{1}}{{3}}{<}{1}$$, so the series converges.
$$\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\left(\frac{{1}}{{3}}\right)}^{{k}}=\frac{{1}}{{{1}–{\left(\frac{{1}}{{3}}\right)}}}=\frac{{1}}{{\frac{{2}}{{3}}}}=\frac{{3}}{{2}}$$
b) $$\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\left(-\frac{{1}}{{2}}\right)}^{{k}}$$
Here, $$\displaystyle{r}=-\frac{{1}}{{2}}$$, so the series converges.
$$\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\left(-\frac{{1}}{{2}}\right)}^{{k}}=\frac{{1}}{{{1}–{\left(-\frac{{1}}{{2}}\right)}}}=\frac{{1}}{{{1}+\frac{{1}}{{2}}}}=\frac{{2}}{{3}}$$

2021-09-22
Look at my photo for the right answer: