Pointwise Convergence of Geometric Series on [0,1)

Yes, the sequence converges pointwise on the set [0,1) to the function f(x) = 1/(1-x).

To see this, note that for any fixed x in [0,1), the series Σn k=0 x^k is a geometric series with first term 1 and common ratio x. By the formula for the sum of a geometric series, we have:

Σn k=0 x^k = 1 + x + x^2 + ... + x^n = (1 - x^(n+1))/(1 - x)

As n approaches infinity, x^(n+1) approaches 0 since x<1. Therefore, we have:

f_n(x) = Σn k=0 x^k → f(x) = 1/(1-x)

This convergence is pointwise since for each fixed x in [0,1), the sequence of partial sums converges to f(x). However, the convergence is not uniform on [0,1) since the rate of convergence depends on x and can be arbitrarily slow as x approaches 1.