Question

How do I find the sum of the infinite series `1/2` + 1 + 2 + 4 +... ?

Answer 1

First of all, don' t hold your breath while counting an INFINITE set of numbers! This infinite Geometric sum has a first term of `1/2` and a common ratio of 2. This means that each successive term is being doubled to get the next term. Adding the first few terms could be done in your head! (perhaps!) `1/2+1= 3/2` and `1/2 + 1 + 2` = 3`1/2`

Now, there is a formula to help you come up with a "Limit" of a sum of terms....but only if the ratio is nonzero. Of course, do you see that adding larger and larger terms will simply make the sum become larger and larger! The guideline is: if |r| > 1, then there is no limit.
If |r| < 1, then the series DIVERGES, or goes toward some particular number value.