Pointwise Convergence of fn(x)=Σx^k on [0,1]

No, the sequence does not converge pointwise on the set [0,1].

To see this, consider the value of fn(x) for x=1. Then we have:

f1(x) = Σx^k = 1 + 1 + 1 + ... (infinitely many terms)

This series diverges, so there is no limit function for f1(x) on the interval [0,1].

In fact, we can see that for any x in [0,1), the series converges to a finite value, given by the formula for a geometric series:

f(x) = Σx^k = 1/(1-x)

But at x=1, the series diverges to infinity. Therefore, the sequence does not converge pointwise on [0,1].