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Find the sum of the series

What is the sum of the series: 2 + 12 + 40 + 112 + .....................infinity ?

Note by Manish Dash
2 years ago

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The series does not converge. Nihar Mahajan · 2 years ago

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@Nihar Mahajan ok so is the answer infinity. @Nihar Mahajan Manish Dash · 2 years ago

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@Manish Dash I guess yes. Nihar Mahajan · 2 years ago

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According to me the answer should be infinity, since the series is increasing and common ratio is greater than 1. But is my reasoning correct? @Brian Charlesworth Manish Dash · 2 years ago

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@Manish Dash Yes, that's correct. The \(n\)th term is of the form \(a(n) = 2^{n}(2n - 1),\) which goes to \(\infty\) as \(n \rightarrow \infty,\) and so by the \(n\)th term test we conclude that the series diverges.

Just out of curiosity, note that the sum of the first \(N\) terms of this series is \((2N - 3)2^{N+1} + 6.\) Brian Charlesworth · 2 years ago

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@Brian Charlesworth Thank You Sir. Manish Dash · 2 years ago

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