Loading [MathJax]/jax/output/HTML-CSS/jax.js
Entry 23
For powers
k=4n+3, Ramanujan gave
∑n=1n3e2nπ−1=13e2π−1+23e4π−1+33e6π−1+⋯=Γ(14)8210⋅5π6−1240 ∑n=1n7e2nπ−1=17e2π−1+27e4π−1+37e6π−1+⋯=3Γ(14)16217⋅5π12−1480 ∞∑n=1n11e2nπ−1=111e2π−1+211e4π−1+311e6π−1+⋯=189Γ(14)24222⋅5⋅13π18−69165520 while for
k=4n+1 it evaluates to a
rational number ∑n=1n5e2nπ−1=15e2π−1+25e4π−1+35e6π−1+⋯=1504 ∑n=1n9e2nπ−1=19e2π−1+29e4π−1+39e6π−1+⋯=1264 ∑n=1n13e2nπ−1=113e2π−1+213e4π−1+313e6π−1+⋯=124 and so on. The case
k=1 diverges.
No comments:
Post a Comment