Monday, 13 May 2013

sequences and series - Prove that sumlimitsin=1nfty(1)nfracx2+nn2 converges uniformly, but not absolutely

My Work:
i) Uniform Convergence (By Weierstrass M-Test):
I am attempting to show that the series converges uniformly on the interval I=[a,a], in which:n=1(1)nx2+nn2n=1(1)na2+nn2=ε0
Suppose that n=1x2+nn2 will be called "A" and ε0 will be called "B". By the M-test, if B converges, then A converges uniformly on the interval I defined above.
I am having difficulty proving that B converges, however. I have tried both the root and ratio tests, and they have been unhelpful. For this segment, could someone confirm my logic / provide a hint towards the convergence of B?
ii) Absolute Convergence:
I believe that this function does not converge absolutely becausen=1x2+nn2 is a divergent sum as n approaches infinity. Bit of a trivial solution here but I believe it's sufficient.

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