MY MATEX NOTES

          Today we will evaluate the following infinite sum which corresponds to the derivative of Dirichlet eta function @ 1 As a bonus we will compute the following integral Lets first introduce a lemma: Lemma 1: (1) Proof: Claim: (2) If we let     in (1) we get (3) Lets now recall the Euler Maclaurin Formula (proved here ) to estimate the last two sums above (4) Choosing    and         in (4), we get for the first sum : (5) And for the second sum (6) Recall also the integral representation of the Stiltjies constant (shown here ): (7) letting in (7) we obtain (8) Now, plugging (5) and (6) back in (3) and letting we obtain We can now use (2) to calculate the following integral Proof:

In this post We will prove the following integral Lets first evaluate the integral (1) Differentiating (1) w.r. to s Now, letting (2) Now recall that (3) And consequently (4) We also have the following relations ( proved here ) and Setting in (3) and (4) we get (5) (6) Plugging (5) and (6) in (2) Appendix Recall the geometric series