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Discuss the convergence or divergence of the series \sum _{{k=0}}^{{\infty }}{\frac  {2}{(k+1)(k+3)}}.

This doesn’t look like a telescopic series but if we do a partial fractions decomposition it will become a telescopic series.

S=\sum _{{k=0}}^{{\infty }}{\frac  {2}{(k+1)(k+3)}}=\sum _{{k=0}}^{{\infty }}\left({\frac  {1}{k+1}}-{\frac  {1}{k+3}}\right)=\sum _{{k=0}}^{{\infty }}\left({\frac  {1}{k+1}}-{\frac  {1}{k+2}}+{\frac  {1}{k+2}}-{\frac  {1}{k+3}}\right)\,

This is basically the sum of two different telescopic series. So we have

S={\frac  {1}{1}}-\lim _{{k\rightarrow \infty }}{\frac  {1}{k+2}}+{\frac  {1}{2}}-\lim _{{k\rightarrow \infty }}{\frac  {1}{k+3}}={\frac  {3}{2}}\,

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