(The question is very loosely related to Pathfinder 2e, so I left out the tag intentionally. It is actually about Trick Magic Item and a Wand of Tailwind)

I need at least a 10 on a d20. I can roll until I get it, or I get a 1. What is the chance I succeed? I would like to have the general formula, not just for 10.

The chance of success on the first try is 55% (11/20). If I do not succeed, which has a chance of 40%, I can try again with the same chance, so should be:

0.55 + 0.4 * 0.55 + (0.4 * 0.55)^2 + ... + (0.4 * 0.55)^n

Generally (tn = target number)

(21 - tn) / 20 + ((tn - 2) / 20) * ((21 - tn) / 20) + ... 

But how do I calculate the sum of this?