Answer:
9 · 2⁹
(-½, ½)
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).