Proof of Partial Sum Error
In the proof of convergence we saw that is monotonically increasing. Since, and every term in brackets is non-positive, we see that is monotonically decreasing. By the previous paragraph, hence . Similarly, since is monotonically increasing and converging to, we have . Hence we have for all n.
Therefore if k is odd we have, and if k is even we have .
Read more about this topic: Alternating Series Test, Proof
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