Question

if x^a.x^b=(x^a)^b prove that a/b +b/a =ab-2​

Answer 1

Answer:

It suffices to prove it for x=e because then xaxb=ealnxeblnx=e(a+b)lnx=xa+b.

eaeb=∑n=0∞ann!∑n=0∞bnn!=∑n=0∞∑k=0nakk!bn−k(n−k)!=∑n=0∞∑k=0n1n!(nk)akbn−k=∑n=0∞(a+b)nn!=ea+b

Here we use Cauchy's product formula, the series converge absolutely.