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

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ANSWERED
asked 2021-09-05
Find the sum of the series, if it converges:
a) \(\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\left(\frac{{1}}{{3}}\right)}^{{k}}\)
b) \(\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\left(-\frac{{1}}{{2}}\right)}^{{k}}\)

Expert Answers (2)

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.
\(\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\left(-\frac{{1}}{{2}}\right)}^{{k}}=\frac{{1}}{{{1}–{\left(-\frac{{1}}{{2}}\right)}}}=\frac{{1}}{{{1}+\frac{{1}}{{2}}}}=\frac{{2}}{{3}}\)

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Best answer
2021-09-22
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