# Calculating sum of infinite series

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?

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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.