My physics textbook tells me that the sum:
(1 + (y/m) + (y/m)^2 + ... + (y/m)^n)
where m < n, equals:
(1 / (1 - y/m)) * (1 - (y/m)^n+1)
...but I can't work out how. Can anyone help?
It's part of a long argument proving the convergence of the power series exp(tX) where X is a 2x2 matrix. They're summing a series (1/m!)y^m + (1/m!m)y^m+1 ... + (1/m!)(1/m^n)y^m+n which (as I can see) equals y^m/m! times the series above.
For $q \ne 1$ the following formula holds:
$$1+q+q^2+...+q^n=\frac{1-q^{n+1}}{1-q}.$$