Question

Compute y' and y"?

Given that y=sigma of n=0, infinity of nxⁿ, compute y' and y" and write out the first four terms of each series as well as the coefficient of xⁿ in the general term.

Answer 1

To compute y' and y'', differentiate the series representation of y term by term. In the first derivative, the coefficients of xⁿ are 0, 1, 4, and 9. In the second derivative, the coefficients of xⁿ are 0, 0, 4, and 18.

Step-by-step explanation:

To compute y' and y'', we first need to differentiate the series representation of y term by term.

For y':

1. Start by differentiating the general term, nxⁿ, with respect to x. This will give us n(n-1)xⁿ⁻¹.
2. Next, multiply the general term by n and sum up the resulting series representation to obtain y'.
3. The first four terms of y' will be 0, 1, 4x, and 9x². The coefficients of xⁿ in the general term are 0, 1, 4, and 9.

For y'':

1. Start by differentiating the general term, n(n-1)xⁿ⁻¹, with respect to x. This will give us n(n-1)(n-2)xⁿ⁻².
2. Next, multiply the general term by n(n-1) and sum up the resulting series representation to obtain y''.
3. The first four terms of y'' will be 0, 0, 4, and 18x. The coefficients of xⁿ in the general term are 0, 0, 4, and 18.