Question:

Is it possible to "calculate":

knowing that:

Answer by Pablo:

This is a very good question, and the answer is not simple. Your question is about infinite series, a really fascinating topic that I couldn't do justice in this short answer. However, I'll give it a try.

The short answer to your question is that it is generally very hard to find the result of an infinite series. Sometimes it is relatively easy, as it is in the case of the sum of the geometric series.

What we could try to do in most cases, though, it is to prove that the limit indeed exists. That's a whole science of its own, and one that can give you a satisfying answer most of the time.

If you really want to find the result of a convergent infinite series, I recommend you to use wolfram alpha. They have a lot of infinite series and it is really easy to use.

Pablo