Please see the attached file for the fully formatted problem.
Define ak recursively by a1 = 1 and
ak = (−1)k
1 + k sin
ak−1, k > 1.
=1 ak converges absolutely.
Since this problem is an analysis problem, please be sure to be rigorous.
Please see the attached file for the complete solution.
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We have to consider the series:
First of all from the given information that a1=1, we see that for k>1:
and then it turns out that:
Absolute convergence is proven for an infinie series of real numbers.