MY MATEX NOTES

             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.

Today, we will proof the following relations: (1) (2) (3) (4) Recall the infinite sine product (5) To proof (2), we recall the relation , so letting in (1) letting       in (1) we obtain Similarly, letting in (2) we obtain Corollary Taking log in both sides of (2) From the relation   

Weierstrass infinite product for the Gamma function I want to derive in this post the Weierstrass infinite product gamma . It is a very useful expression to solve many problems involving the gamma function. 1.Weierstrass infinite product 1. The Weierstrass infinite product is given by Recall Gauss infinite product for the Gamma function derived in this post   (1.1) Now, and finally taking the limit! (1.2) or And recalling the functional equation  of the Gamma Function we get (1.3) 2. Series Expansion Digamma function If we take logs of (1.3) and differentiate with respect to x we get the Series representation of the digamma function , extremely useful (2.1) 3. Series expansion Trigamma Function and the evaluation of an Infinite series If we differentiate (2.1) we get (3.1) if we let and we get that putting all together (3.2) On the other hand we have the reflection formula for the Gamma function taking log on both sides and differenti...