how to solve
If S= integral of n*x^(n-1)/(1+x) for n>=1 with limit 0 to 1
then find limit of sequence {S} if n tends to infinity.
i had made a mistake. its 1/2
here's my solution,
Now, our aim is to get rid of the n in the first place...
i used by parts for that.
using by parts taking 1/(1+x) as the first function, we get;
The first term of the sum has the limits 0->1 & we get rid of n from both terms by multiplying the n outside...
So,
Now, applying the limits of integration in the first terms gives us,
Then,
Now, observe the second term, here, x lies between 0 & 1. so x is basically a fraction & if we increase the power of a fraction, it decreses.
If we take the power to infinity, it almost (tends to) 0.
Hence, we have,