MY MATEX NOTES

We have previously showed that Today we will revisit this remarkable result proving it by a different method, namely, we will show that And compute this integral. First, lets find a series expansion for   : Letting we obtain (1) Now recall (see here ) (2) Letting we obtain (3) Claim: Proof: Letting k=3  we obtain Let´s now calculate this integral Where we used (see here )

          In this post We will derive the remarkable beatiful Hjortnes series used by Apery to prove the irrationality of zeta(3) Recall (proved here ) (1) Letting in (1) and using the fact that    we obtain (2) Proof: Dividing (2) by x and integrating from 0 to 1/2 we obtain (3) Let´s now focus on the R.H.S. of (3) Plugging the result of the integral back in (3) we obtain the remarkable result Appendix 1 Recall the following relations regarding the Golden Ratio (A.1) We have proven the following relation in this post (A.2) Also, recall the Trilogarithm identity proved here (A.3) And the Polylogarithm relation proved here (A.4) Example, letting in (A.4) we obtain (A.5) Claim: (A.6) Proof: If we let in (1) we obtain Appendix 2 And we get (A.7) Now we focus on the integral on the L.H.S. Plugging this result back in (A.7) we obtain (A.8) Example, letting x=1 in (A.8) we obtain Reference POLYLOGAR...

Easy looking integral, not so easy… I saw this post from @infseriesbot and wanted to proof at least the first one. After a hardwork to compute this easy looking integral, I found out that the result stated by @infseriesbot is incorrect! Exctly, this time @infseriesbot is wrong. Checking numerically confirms the computation here. Proof let Now use the fact that for The second integral is a little trickier Observe the following dividing both sides by and integrating from to Now we have to evaluate the three integrals on the RHS and we are done. I´ll state the value of each of the integrals and show the proof in the end. The next integral is the same as so no need to proof it again. Putting all together we get Now summing the results of and we get the final result!   Checking numerically we have   and  , which agrees with WolframAlpha ! A Corollary We just computed the integral On the other hand we can show that this inte...