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For each of the following series,
decide if it converges or diverges,
and if it converges compute its value.
\[
\frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \frac{16}{81} + \dotsb
\qquad\qquad
-\frac{2}{3} + \frac{4}{9} - \frac{8}{27} + \frac{16}{81} - \dotsb
\]
\[
\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - \frac{16}{81} + \dotsb
\qquad\qquad
\frac{10}{3} + \frac{20}{9} + \frac{40}{27} + \frac{80}{81} + \dotsb
\]
\[
\frac{3}{2} + \frac{9}{4} + \frac{27}{8} + \frac{81}{16} + \dotsb
\]
\[
\sum_{n=1}^\infty \frac{2n^3}{7n^3+6n}
\qquad
\sum_{n=1}^\infty 3^{n}\,10^{-n}
\qquad
\sum_{n=1}^\infty 3^{2n}\,10^{2-n}
\qquad
\sum_{n=1}^\infty \frac{4}{n(n+4)}
\]
\[
\sum_{n=1}^\infty \arctan(n)
\qquad
\sum_{n=1}^\infty \frac{\mathrm{e}^n}{n^\mathrm{e}}
\qquad
\sum_{n=1}^\infty \ln\left(\frac{n}{n+2}\right)
\]