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Calculus/Calculus - Find the limit

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Expert: - 10/16/2008


Question
This one is kicking my butt, I feel really dumb.  How do I find the the limit.
lim   ∑ (1 + (i / n)) (2 / n)
n→∞   i =1


Answer
I don't think it will kick too hard...

Over i is the sum from 1 to ∞.

The summation can be changed into 2Σ(over i)(1/n + i/n).  

It is known that 2Σ(over i)1/n is the limit(n->∞)n/n.  
The n's cancel, leaving 1.

It is know that 2(Σi)/n is 2(n(n+1)2)/n, with the 2's and the n's cancelling, so that you have n+1.  Since n is going to ∞, the summation is going to ∞.

Note that the first term was convergent, but the second term was not.
Since this is the case, and their both positive and one of them is divergent, we can say that when added together they are divergent.

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