I got my formual from a Handbook of Tables and Formulae that I keep handy. I have a small but good collection of technical books. This formula is referred to as the formula for an annuity whose present value is one. You then multiply by the amount.

I suspect that there is a way to sum your formula to arrive at mine.

Many years ago I saw my formula derived as the sum of a series. If summing your series arrives at the same result, I would bet money that there is a way to derive my formula from yours. Your series looks like an arithmetic progression, for which there is some simple formula.

My HP calculator has financial functions built in. I used it for the Immediate payment calculation, for which I do not know the precise forumla.

I find it hard to believe that a bank does not have mortgage payment tables and formulae available. That is what we are dealing with here.