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    Algebra: Convergence of Series

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    Please help with the following problem. This problem is from Advanced Calculus II class. It is also an introduction to real analysis.

    Consider the series n = 1 to infinity 1/( 1 + n^2 x)

    (a) For what values of x in R does the series converge absolutely?

    (b) On what intervals of R does it converge uniformly?

    (c) On what intervals of R does it fail to converge uniformly?

    (d) Is the series bounded on R?

    Please I want detailed answers with full explanation..after all I want to learn not just have the right answer. I appreciate your help. Thanks.

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    Solution Preview

    a) For what values of x in R does the series converge absolutely?

    Case 1. Let x= 0.
    Then the series equals n so it is not convergent, abs converges or uniformly convergent.

    Case 2.
    Let an = 1/( 1 + n^2 x).
    Now Σ|an| converges.
    Now Σ|an| < Σ 1/|n²x| = 1/|x| Σ 1/n² which is ...

    Solution Summary

    This posting helps with basic algebra problems. The calculations include values, intervals, uniform convergence. Step by step calculations are given for each problem.