Thursday, 30 April 2015

calculus - Evaluate sumlimitsinftyn=0sumlimitsnr=0left(frac1(nr)!anrright)left(frac1r!brright)



I'd like to Prove that n=0nr=0(1(nr)!anr)(1r!br)=(n=01n!an)(n=01n!bn)




I do as follow




n=0nr=0(1(nr)!anr)(1r!br)=0r=0(1(0r)!a0r)(1r!br)+1r=0(1(1r)!a1r)(1r!br)+2r=0(1(2r)!a2r)(1r!br)+




I couldn't able to get the right hand



Any help will be appreciated! Thanks



Answer



You might find it easier to start from the RHS and show that



(n=01n!an)(n=01n!bn)=n=0nr=0(1(nr)!anr)(1r!br).



Actually, for me this is the only step. It's just how you multiply two series.



Something more interesting would be to see what you can make of



1(nr)!r!anrbr=1n!(nr)anrbr




and the Binomial Theorem.


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