Pointwise Convergence of Geometric Series on [0, 1]

Yes, the sequence fn(x) = Σn k=0 x^k 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 sequence fn(x) is a geometric series with first term 1 and common ratio x. Therefore,

fn(x) = 1 + x + x^2 + ... + x^n = (1 - x^(n+1))/(1 - x)

As n approaches infinity, x^(n+1) approaches zero, so fn(x) approaches 1/(1-x). Therefore, the limit function is f(x) = 1/(1-x).