I'd like to Prove that ∞∑n=0n∑r=0(1(n−r)!an−r)(1r!br)=(∞∑n=01n!an)(∞∑n=01n!bn)
I do as follow
∞∑n=0n∑r=0(1(n−r)!an−r)(1r!br)=0∑r=0(1(0−r)!a0−r)(1r!br)+1∑r=0(1(1−r)!a1−r)(1r!br)+2∑r=0(1(2−r)!a2−r)(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=0n∑r=0(1(n−r)!an−r)(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(n−r)!r!an−rbr=1n!(nr)an−rbr
and the Binomial Theorem.
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