Sequence from generating function with integral

Question

So, let $A(x)$ be the generating function of $a_0,a_1,\dots$ then what would be the sequence of the generating function: $$\int^x_0 A(t)dt$$

Since I am not much acquainted with integrals any help would be much appreciated.

Answer 1

The sequence for $A(x)$ is $a_0, a_1, a_2, \cdots, a_n, \cdots$ $$A(x) = a_0x^0 + a_1x^1 + a_2x^2 + \cdots + a_n x^n +\cdots$$ $$\int_0^x{A(t)} = a_0\frac{x^1}{1} + a_1\frac{x^2}{2} + a_2\frac{x^3}{3} + \cdots a_{n-1}\frac{x^n}{n}+\cdots$$ So the sequence for the integral of $A(x)$ is: $$0, \frac{a_0}{1}, \frac{a_1}{2}, \frac{a_2}{3}, \cdots, \frac{a_n}{n+1}, \cdots$$