I have the following series: \sum_{n=1}^\infty(2^{\frac{1}{n}}+2^{-\frac{1}{n}}-2)



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I have the following series:

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Beginner2022-03-01Added 6 answers

Note that ex+ex2=x2+x412+x6360++2x2n(2n)!+
It's easy to show that this series has a bound of the form ex+ex2<kx2, where k is some positive constant, for all x sufficiently small. We could use the Taylor remainder theorem or, more simply, just compare the series to the clearly greater geometric series x2+x4+x6+x8+=x21x2, which gives us the bound (for example)
ex+ex2<43x2 valid for |x|<12
Your sum is just adding the values of this expression at x=ln2n, so it has the same convergence properties as n=11n2, which converges.

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