Answer
$$ \sum^{\infty}_{n=0}(n + 1)a_nx^n$$
Work Step by Step
1. Write the x inside the series on the first expression:
$$\sum^{\infty}_{n=1} na_nx^{n-1}x = \sum^{\infty}_{n=1} na_nx^{n}$$
2. Find the term where n = 0 for that expression:
$$(0)a_0x^{-1} = 0$$
Therefore:
$$ \sum^{\infty}_{n=0} na_nx^{n} = 0 + \sum^{\infty}_{n=1} na_nx^{n}$$
$$ \sum^{\infty}_{n=0} na_nx^{n} = \sum^{\infty}_{n=1} na_nx^{n}$$
3. Now that both expressions start at the same integer (0), we can write them as one series:
$$\sum^{\infty}_{n=0}(na_nx^n + a_nx^n) = \sum^{\infty}_{n=0}(na_n+ a_n)x^n = \sum^{\infty}_{n=0}(n + 1)a_nx^n$$
