You can make up yourself an infinite geometric series. Just pick any number that you want as your 1st. term and then multiply it by ratio greater than 1 and it will diverge...........etc.
Example: Your 1st. term=1 and the ratio is 2, then you would have:
1, 2, 4, 8, 16, 32, 64......... and so on for as long as you want. So, by definition, you cannot sum up an infinite geometric that diverges. But, you can sum up an infinite series that converges such as this one: 1 + 1/2 + 1/4 + 1/8 + 1/16..................forever =2!!.
F x {[1 - r^n] / [1 - r]}=Sum of a geometric series, where F=1st. term, r=ratio, n=number of terms.
Using this formula, you can sum up, say, the 1st. 100 terms of the above series: 1, 2, 4, 8.......etc. and you will get=1.27 x 10^30.