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Test the convergence

\[\large{\displaystyle \sum_{n=1}^{\infty} x^n}\]

if \(x>1\)

Note by Tanishq Varshney
1 year, 9 months ago

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@Tanishq Varshney Well, is it convergent? Satyajit Mohanty · 1 year, 9 months ago

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@Satyajit Mohanty dont know , was given homework Tanishq Varshney · 1 year, 9 months ago

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@Tanishq Varshney Where do you study? Undergraduate or High School or Senior Secondary (+2) ? Satyajit Mohanty · 1 year, 9 months ago

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@Satyajit Mohanty under graduate, btw thanks for helping, are u sure it diverges Tanishq Varshney · 1 year, 9 months ago

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@Tanishq Varshney Well Hmm, yup it's divergent! But anything can happen due to analytic continuations :D

For example \(\zeta(-2n) = 0\) \(\Rightarrow 1^{2n} + 2^{2n} + 3^{2n} + 4^{2n} + \ldots = 0\) for \(n \in \mathbb {Z}^+\)

I just know that:

\(\large{\displaystyle \sum_{n=1}^{\infty} ar^n}\) converges only when \(|r| < 1\), \(a \neq 0\). It diverges for \(|r| \geq 1\). Satyajit Mohanty · 1 year, 9 months ago

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@Satyajit Mohanty one more doubt is \(\displaystyle \sum_{r=1}^{\infty} r\) convergant or divergant. Becoz ramanujan proved this equal to -1/12 Tanishq Varshney · 1 year, 9 months ago

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@Tanishq Varshney \(\displaystyle \sum_{r=1}^\infty r \) is divergent. But again, Analytic Continuation changes the game and we find it as \(-\frac{1}{12}\). Satyajit Mohanty · 1 year, 9 months ago

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