Re: Integration anomaly?

*To*: mathgroup at smc.vnet.net*Subject*: [mg127021] Re: Integration anomaly?*From*: Bill Rowe <readnews at sbcglobal.net>*Date*: Sun, 24 Jun 2012 04:26:50 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

On 6/23/12 at 4:16 AM, john_szumiloski at merck.com (Szumiloski, John) wrote: >I recently was playing around with the function Log[1 + 1 / ( t^n ) >], and exploring positive values of n. (I have no interest in >nonpositive or complex n) In particular, I wanted to look at its >integral, so I did this: (v8.0.4, Windows XP) >Integrate[ Log[1+1/(t^n)], { t, 0, Infinity } ] >which gave: > >ConditionalExpression[ -(Pi Csc[Pi/n]), Re[n]<0 ] I get the same result using In[7]:= $Version Out[7]= 8.0 for Mac OS X x86 (64-bit) (October 5, 2011) >Now I am no analysis expert, but it seems pretty clear to me that >the integral diverges for negative (real) n. Not proof but given In[8]:= Table[NIntegrate[Log[1 + 1/(t^n)], {t, 0, Infinity}], {n, 5}] Out[8]= {23233.1,3.14159,3.6276,4.44288,5.3448} and In[9]:= Table[NIntegrate[Log[1 + 1/(t^-n)], {t, 0, Infinity}], {n, 5}] Out[9]= {5.522502947974294*10^27954,1.104500589594859*10^27955,1.656750884392288*10^27955,2.209001179189718*10^27955,2.761251473987147*10^27955} it seems clear the integral converges for n<0 and diverges for n>0