# Integral of Generating Function

## Theorem

Let $\map G z$ be the generating function for the sequence $\sequence {a_n}$.

Then:

 $\ds \int_0^z \map G t \rd t$ $=$ $\ds \sum_{k \mathop \ge 1} \dfrac {a_{k - 1} z^k} k$ $\ds$ $=$ $\ds a_0 z + \dfrac {a_1 z^2} 2 + \dfrac {a_2 z^3} 3 + \dfrac {a_3 z^4} 4 + \cdots$

## Proof

 $\ds \int_0^z \map G t \rd t$ $=$ $\ds \int_0^z \paren {\sum_{k \mathop \ge 0} a_k t^k} \rd t$ Definition of Generating Function $\ds$ $=$ $\ds \sum_{k \mathop \ge 0} \paren {\int_0^z a_k t^k \rd t}$ $\ds$ $=$ $\ds \sum_{k \mathop \ge 0} \paren {\intlimits {a_k \dfrac {t^{k + 1} } {k + 1} } 0 z}$ Primitive of Power $\ds$ $=$ $\ds \sum_{k \mathop \ge 0} \paren {a_k \dfrac {z^{k + 1} } {k + 1} }$ $\ds$ $=$ $\ds \sum_{k \mathop \ge 1} \dfrac {a_{k - 1} z^k} k$ Translation of Index Variable of Summation

$\blacksquare$