# Algebra: Convergence of Series

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 Î£|an| converges.

Now Î£|an| < Î£ 1/|nÂ²x| = 1/|x| Î£ 1/nÂ² 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.