Question

What do I do to implement the `x^2` into this series? `x^2sum_(n=0)^oo(na_nx^(n-1))`

Answer 1

`sum_(n=0)^oo (na_nx^(n+1))`

Explanation:

Let:
`S = x^2sum_(n=0)^oo(na_nx^(n-1))`

If unclear as to the effect then the best option to expand a few terms of the summation:

`S = x^2{0a_0x^(-1) + 1a_1x^0 + 2a_2x^1 + 3a_3x^2 + 4a_4x^3 + ...}`
`\ \ = {0a_0x^(1) + 1a_1x^2 + 2a_2x^3 + 3a_3x^4 + 4a_4x^5 + ...}`

Then we can put it the series back into "sigma" notation:

`S = sum_(n=0)^oo (na_nx^(n+1))`