
proof of raabe criteria.
does someone know how to prove raabe's criteria for sums convergence?
i tried to prove it with a hint of the book but i got for example if we get from planetmath website, i got that mu is smaller than 1.
because when the sum is convergent:
(1a_n+1/a_n)*n=1>=mu
i had a hint from my book that i need to prove that {na_n+1} is monotonely decreasing i.e for every n na_n+1>(n1)a_n and then i only need to prove that c_n=(n1)a_nna_n+1 is a convergent sum, which i did but i didn't get the ineality i needed.
any help is appreciated.
here's the link from planetmath:
http://planetmath.org/encyclopedia/RaabesCriteria.html

Check what this implies for $\displaystyle {\rm limsup}\frac{a_{n+1}}{a_n}$ and use the ratio test.