Finding the interval of convergence of the integral

Question

Let f(x)= ∑n=1∞ (-1)^n (x/4)^n. Find the series and interval of convergence for the integral from 0 to x of f(t)dt

Answer 1

This is an alternating geometric series with r = - x/4. A geometric series converges iff | r | < 1.

Thus, the interval of convergence is -4 < x < 4.

When | r | < 1, S_∞ = a₁ / (1 - r).

Here a₁ = -x/4 and r = -x/4 so the sum, in terms of x, = -x/4 / (1 + x/4) = - x / (4 + x).