Calc6.33

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Discuss the convergence or divergence of the series \sum _{{n=10}}^{{\infty }}\left({\frac  {3}{5}}\right)^{n}.

This series converges since the common ratio is between 0 and 1. To do the sum, let us make use of the formula for the finite geometric sum.

\sum _{{n=10}}^{{\infty }}\left({\frac  {3}{5}}\right)^{n}=\sum _{{n=0}}^{{\infty }}\left({\frac  {3}{5}}\right)^{n}-\sum _{{n=0}}^{{9}}\left({\frac  {3}{5}}\right)^{n}={\frac  {1}{1-{\frac  {3}{5}}}}-{\frac  {1-\left({\frac  {3}{5}}\right)^{{10}}}{1-{\frac  {3}{5}}}}={\frac  {\left({\frac  {3}{5}}\right)^{{10}}}{{\frac  {2}{5}}}}={\frac  {59049}{3906250}}\,


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