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Infinite Series of Real Numbers (Absolute Convergence)

Please see the attached file for the fully formatted problems.

Suppose ak  0 and a1/k
k ! a as k ! 1. Prove that
=1 akxk converges
absolutely for all |x| < 1/a if a 6= 0 and for all x 2 R if a = 0.
Since this problem is an analysis problem, please be sure to be rigorous.


Solution Summary

Absolute convergence is proven for an infinite series of real numbers. The solution is detailed and well presented.