Wednesday, 21 October 2015

real analysis - sumlimitsin=1nftylog(1+an) converges absolutely iffsumlimitsin=1nftyan converges absolutely.




n=1log(1+an) converges absolutelyn=1an converges absolutely.





How to prove this,



Suppose n=1an converges absolutely.

Let un=an and vn=log(1+an), then limnunvn=1>0n=1log(1+an) converges absolutely.
How to prove the converse part?


Answer



Hint: From the definition of ln(1), we have



limu0ln(1+u)u=1.



Thus there is a>0 such that




12|ln(1+u)u|32



for u(a,a),u0.


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