$f(x) = \sum_{n=1}^{\infty} n x^{n-1}$ with $R = 1$

$f(x) = \sum_{n=0}^{\infty} (n+1) x^n$ with $R = 1$

$f(x) = \sum_{n=2}^{\infty} n(n-1) x^{n-2}$ with $R = 1$

The integral $\int_0^{1/2} f(x)\,dx = \sum_{n=1}^{\infty} \frac{1}{(n+1)2^{n+1}}$