Infinite Series of Real Numbers (Absolute Convergence)
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Define ak recursively by a1 = 1 and
ak = (−1)k
1 + k sin
1
k
−1
ak−1, k > 1.
Prove that
P
1k
=1 ak converges absolutely.
Since this problem is an analysis problem, please be sure to be rigorous.
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Solution Summary
Absolute convergence is proven for an infinie series of real numbers.
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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:
...
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