:::::::::::::Please, the less you say the phrase "infinite sum", the happier you'll be. Obviously no one can compute an infinite sum. But the sum of an infinite series is defined as the limit of the sequence of its partial sums. No, this definition does not follow from any earlier definitions; that why we need to make it a definition in the first place! Hundreds of years ago, mathematicians argued over infinite series, and some of them arrived at comically bad results. In hindsight, the confusion could have been avoided if they'd bothered to define what they were talking about. A good definition doesn't have to be intuitively obvious; it just has to be useful. There is an extensive body or research on infinite series and their sums... and it all works.