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.© BrainMass Inc. brainmass.com May 24, 2023, 1:55 pm ad1c9bdddf
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.
Let an = 1/( 1 + n^2 x).
Now Σ|an| converges.
Now Σ|an| < Σ 1/|n²x| = 1/|x| Σ 1/n² which is ...
This posting helps with basic algebra problems. The calculations include values, intervals, uniform convergence. Step by step calculations are given for each problem.