Combining Infinite Series: Can I Make These Two Series Start at the Same Point?

[Jamin2112]

Homework Statement

Rewrite the given expression as a sum whose generic term involves xⁿ

[m=2 to ∞] ∑m(m-1)a_mx^m-2 + [k=1 to ∞] x∑ka_kx^k-1

Homework Equations

None in this problem

The Attempt at a Solution

To make the first part involve only xⁿ, I can use the substitution n=m-2.

[n=0 to ∞] ∑(n+2)(n+1)a_n+2xⁿ.

But I can't make the second part in terms of xⁿ and [n=0 to ∞], as far as I know.

[k=1 to ∞] x∑ka_kx^k-1 = [k=1 to ∞] ∑ka_kx^k = [n=1 to ∞] ∑na_nxⁿ

If I try to take change the start of the sum to n=0, that will effect the xⁿ. See what I'm saying? I want to combine this into one big sum from n=0 to ∞.

 

[D H]

[k=1 to ∞] x∑ka_kx^k-1 = [k=1 to ∞] ∑ka_kx^k = [n=1 to ∞] ∑na_nxⁿ

If I try to take change the start of the sum to n=0, that will effect the xⁿ. See what I'm saying?

So don't do that then. At this stage you have two series, both of which are in terms of xⁿ. The only problem is one starts at n=0 while the other starts at n=1. Can you make them start at the same point?