Find set on which the series $\sum \frac{1}{n(x^2+n)}$ converge uniform. My solution is as follows $|1/(x^2+n)|≤1$ so that $|1/n(x^2+n)|\leq1/n$. Since $\sum1/n$ converges to zero as n goes to infinity , then by Weierstrass test the series converges uniform. Am I in the right track?, I don’t know how I can get values of $x$ for which the given series is uniform convergent. Thanks for any kind of help.
Uniform convergence of $\sum \frac{1}{n(x^2+n)}$
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real-analysis
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0Just taking a look at $\zeta(2)$ would also help. – 2012-01-07
1 Answers
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Hint: The estimate ${1\over x^2+n}\le 1$ is "too much"; you are throwing away a term that actually helps you (the $n$). Estimate with ${1\over x^2+n}\le {1\over n}$ instead.