Partial sums
Task number: 3869
Verify that if \(a(x)\) is the generating function for the sequence \((a_0,a_1,a_2,\ldots)\), then \(\frac{a(x)}{1-x}\) is the generating function for the sequence of partial sums \((a_0,a_0+a_1,a_0+a_1+a_2,\ldots)\).
Solution
The function \(\frac1{1-x}\) is the generating function for the sequence of ones.
The product of functions \(a(x)\) and \(\frac1{1-x}\) corresponds to the sequence, whose \(n\)-th element is equal to \(\sum_{i=0}^n a_i\cdot 1=a_0+a_1+…+a_n\).
