MY MATEX NOTES

In this post we will prove the following nice series representation for : Click here for the proof. We used the previous established result Click here for the proof.

       Today We will compute the following integral following the same ideas of the previous post : Recall (see here ) (1) Integrating both sides of (1)  from 0 to Computing    Recall (see here ) Letting we obtain We are looking for the Imaginary part of the equation above: Computing the quantities: The Glaisher function We know that (see here ): If we integrate from 0 to x we obtain Integrating from 0 to x we obtain

             In today´s post We will prove the following two alternating infinite sums involving the reciprocal of the central binomial coefficient that appear in this Twitter post First recall previously proved here : (1) Dividing both sides of (1) by x we obtain (2) Now let in (2) (3) If we integrate both sides of the above equation from 0 to 1 we obtain (4) Let´s now evaluate the integral on the R.H.S. We used the fact that and and that Now plugging the result obtained back in (4) we conclude that Proving the first series. For the second one, we integrate (3) from 0 to 1/2 to get Let´s now evaluate the integral on the R.H.S. And we prove the second series. In the evaluation of the last integral we used that: and Proved previously in this post . Also for the integral Appendix: Proof: Let Then On the other hand Equating (A.1) and (A.2) we obtain the desired result.

In today´s post we will compute the amazing alternate infinite sum Where in (*) we used the generating function previously proved here Evaluation of   Evaluation of Where we used the following result Evaluation of Where We used the previously proved results Evaluation of Appendix Proof: We start from On the other hand Equating both equations we obtain the desired result