How to determine if a series has a sum?

You can use sigma notation to represent an infinite series. For example, ∞∑n=110 (12) n-1 is an infinite series. The infinity symbol that placed above the sigma notation indicates that the series is infinite. To find the sum of the above infinite geometric series, first check if the sum exists by using the value of r.

The only general answer is to say that by checking that as more terms of the series are added, the partial sum of all those terms converges. It does not need to have a geometric convergence, so unfortunately you can't use the ratio os the geometric series. If your series were only geometric, then yes, if |r|<1. But in general, only if the partial sums are convergent, that is if the more terms you add, the result changes closer and closer to zero.

It can be written mathematically, such that there exists a term k for which the sum from k to infinity is bounded, but you may not need all the jargon.

Again, make sure there was not an assumption about a specific type of series, like geometric, which is the one considered in the other answer