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  • The value of Lim 1/ n ^k [ (n+1)^k (n+2)^k.......(n+n)^k]^ 1/n As x --> infinity

The value of Lim 1/ n ^k [ (n+1)^k (n+2)^k.......(n+n)^k]^ 1/n As x --> infinity

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Solution: Given  ,

                     lim_x o inftyfrac1n^k[(n+1)^k(n+2)^k(n+3)^k.........(n+n)^k]^frac1n

              Limit be equal  to y

               \ \ Rightarrow log y=frac1nlim_x o infty[log (fracn+1n)^k+log (fracn+2n)^k+........]\ \ Rightarrow log y=lim_x o inftysum [log (fracn+rn)^k]\ \ Rightarrow log y=frac1nlim_x o inftysum_r=1^nklog (1+fracrn)=k int_0^1log (1+x)dx\ \ Rightarrow kcdot 2[log 2 -frac12]=k[log 4 -log e]=klog (frac4e)

  Hence ,             log y=log (frac4e)^kRightarrow y=(frac4e)^k

  

        

        

Posted by

Deependra Verma

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