Tell whether the series converges or diverges, and find sum, if it converges.

Wierzycaz 2021-08-19 Answered
Tell whether the series converges or diverges, and find sum, if it converges.
\(\displaystyle{\sum_{{{n}={3}}}^{\infty}}{\left(\frac{{1}}{{{n}–{2}}}–\frac{{1}}{{n}}\right)}\)

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Expert Answer

averes8
Answered 2021-08-20 Author has 6142 answers

\(\displaystyle{\sum_{{{n}={3}}}^{\infty}}{\left(\frac{{1}}{{{n}–{2}}}–\frac{{1}}{{n}}\right)}\)
\(\displaystyle{\sum_{{{n}={3}}}^{\infty}}\frac{{1}}{{{n}\times{\left({n}-{2}\right)}}}\)
This is not geometric series
\(\displaystyle{S}_{{n}}={1}+\frac{{1}}{{2}}=\frac{{3}}{{2}}\ {a}{s}\ {n}\to\infty.\)
Thus, series converges.

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