Tell whether the series converge. If it converges, find the sum.

Jason Farmer 2021-08-19 Answered
Tell whether the series converge. If it converges, find the sum.
1) $$\displaystyle{\sum_{{{n}={0}}}^{\infty}}{\left(\frac{\pi}{{2}}\right)}^{{n}}$$
2) $$\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\left(\frac{{3}}{{7}}\right)}^{{n}}$$

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Caren

1) $$\displaystyle{\sum_{{{n}={0}}}^{\infty}}{\left(\frac{\pi}{{2}}\right)}^{{n}}$$
$$\displaystyle{a}={1}=\frac{{\left(\frac{\pi}{{2}}\right)}^{{2}}}{{\frac{\pi}{{2}}}}=\frac{\pi}{{2}}={1.57}$$
Thus, the series diverges.
2) $$\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\left(\frac{{3}}{{7}}\right)}^{{n}}$$
$$\displaystyle{a}=\frac{{3}}{{7}}{<}{1}$$, so the series converges.
Sum: $$\displaystyle\frac{{\frac{{3}}{{7}}}}{{{1}–\frac{{3}}{{7}}}}=$$
$$\displaystyle=\frac{{3}}{{7}}\times\frac{{7}}{{4}}=\frac{{3}}{{4}}$$