Sunday, October 9, 2016

real analysis - prove log(1+x)0

Prove log(1+x)<x for x>0


my attempt:


I show ex>1+x for x>0 since


ex=1+x+n(n1)2x2n2+...


so if x>0 then all terms are positive


so ex>1+x for x>0


now given ex>1+x for x>0, can I take log on both sides and show


log(1+x)<x for x>0


or do I have to prove firstly that exp(x)=ex for x>0 then I can take log..

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