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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, 8 months ago

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The series does not converge.

- 2 years, 8 months ago

ok so is the answer infinity. @Nihar Mahajan

- 2 years, 8 months ago

I guess yes.

- 2 years, 8 months ago

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

- 2 years, 8 months ago

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 $$(2N - 3)2^{N+1} + 6.$$

- 2 years, 8 months ago

Thank You Sir.

- 2 years, 8 months ago