Question

How do you do this question?

Answer 1

Answer:

∑ₙ₌₀°° (-3x)ⁿ / n!

R = ∞

Step-by-step explanation:

Maclaurin series is:

∑ₙ₌₀°° f⁽ⁿ⁾(0) xⁿ / n!

Find the nth derivative and evaluate at x=0:

f(x) = e⁻³ˣ

f⁽ⁿ⁾(x) = (-3)ⁿ e⁻³ˣ

f⁽ⁿ⁾(0) = (-3)ⁿ

The Maclaurin series is therefore:

∑ₙ₌₀°° (-3)ⁿ xⁿ / n!

∑ₙ₌₀°° (-3x)ⁿ / n!

Use ratio test to find the radius of convergence.

lim(n→∞)│aₙ₊₁ / aₙ│< 1

lim(n→∞)│[(-3x)ⁿ⁺¹ / (n+1)!] / [(-3x)ⁿ / n!]│< 1

lim(n→∞)│(-3x) n! / (n+1)!│< 1

lim(n→∞)│3x / (n+1)│< 1

0 < 1

The radius of convergence is infinite.