How do you do this question?

9 · 2⁹

(-½, ½)

Step-by-step explanation:

The nth term of the original power series is 2ⁿ xⁿ.

Therefore, the nth term of the derived power series is:

d/dx (2ⁿ xⁿ) = n · 2ⁿ xⁿ⁻¹

When n=9, the term is 9 · 2⁹ x⁸.

Use ratio test:

lim(n→∞)│aₙ₊₁ / aₙ│

lim(n→∞)│((n+1) · 2ⁿ⁺¹ xⁿ) / (n · 2ⁿ xⁿ⁻¹)│

lim(n→∞)│(n+1) · 2x / n│

2│x│

The series converges when the limit is less than 1.

2│x│< 1

│x│< ½

-½ < x < ½

Check the endpoints.

If x = -½, the series is ∑ n · 2ⁿ (-½)ⁿ⁻¹ = ∑ -2n · (-1)ⁿ, which diverges.

If x = ½, the series is ∑ n · 2ⁿ (½)ⁿ⁻¹ = ∑ 2n, which diverges.

So the interval of convergence is (-½, ½).