Partial sum of a series (help!)

[Feynmanfan]

Hello everybody!

I'm having some trouble with series. My calculus teacher asked us to find the partial sum of

Sigma from 1 to n [n^-(1 + 1/n)]

It is obvious that the series diverges when trying to find the infinite sum. However, is it possible to find an expression dependant of n of the partial sum? I don't know where to start from

[HallsofIvy]

Well, one problem you have is that your problem doesn't make sense.

You shouldn't use "n" for both the upper limit of summation and as the index inside the sum.

I assume that what you really mean is
\Sigma_{i=1}^n i^{1+\frac{1}{i}} .

There is no simple formula so you can't just plug a number in.

I recommend that you try some values of n and see what happens:

If n= 1, the sum is simply 1^{1+1}= 1
If n= 2, the sum is 1+ 2^{1+1/2}= 1+ 2\sqrt{2}
If n= 3, the sum is 1+ 2\sqrt{2}+ 3^{1+ 1/3}= 1+ 2\sqrt{2}+ 3(3)^{\frac{1}{3}}

I see a pattern but I don't see any simple way of writing that.