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# A telescoping arctan series

Evaluate the sum

Solution

First of all note that:

Hence letting we have that

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1. Interesting remark: Let us consider the function

then we note that

2. Other interesting serieses of this form are:

1. .

Proof: We have successively:

2. .

Proof: We simply note that

and

Thus, the series telescopes.

3. .

Proof: The same as before note that the sum telescopes , since:

The result follows.

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