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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, 5 months ago

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The series does not converge.

Nihar Mahajan - 2 years, 5 months ago

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ok so is the answer infinity. @Nihar Mahajan

Manish Dash - 2 years, 5 months ago

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I guess yes.

Nihar Mahajan - 2 years, 5 months 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, 5 months ago

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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, 5 months ago

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Thank You Sir.

Manish Dash - 2 years, 5 months ago

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