by Sam
Last Updated September 21, 2018 10:20 AM

I am really confused by sum of infinite series and was hoping to get some guidance. If I have a series $$$\sum_{i=1}^\infty \frac{i}{i+1}$$

If I am asked to calculate the sum of the series, do I just take the limit?

If I take the limit it is 1, which I think means that series diverges so there is no sum.

Or is it the case that the sum is 1? Does a series only diverge if the limit is infinity or does it diverge if the limit doesn't equal 0?

It means that the sum diverges. You are summing numbers that are larger than $\frac{1}{2}$, meaning that the sum of the first $n$ terms will be at least $\frac{n}{2}$. That is, the partial sums grow without bound. As the sum of an infinite series is defined to be the limit of the partial sums, the series diverges.

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