How do you do this question?

∑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.

simplifying

5(x + -6) = 2(x + 3)

reorder the terms:

5(-6 + x) = 2(x + 3)

(-6 * 5 + x * 5) = 2(x + 3)

(-30 + 5x) = 2(x + 3)

reorder the terms:

-30 + 5x = 2(3 + x)

-30 + 5x = (3 * 2 + x * 2)

-30 + 5x = (6 + 2x)

solving

-30 + 5x = 6 + 2x

solving for variable 'x'.

move all terms containing x to the left, all other terms to the right.

add '-2x' to each side of the equation.

-30 + 5x + -2x = 6 + 2x + -2x

combine like terms: 5x + -2x = 3x

-30 + 3x = 6 + 2x + -2x

combine like terms: 2x + -2x = 0

-30 + 3x = 6 + 0

-30 + 3x = 6

add '30' to each side of the equation.

-30 + 30 + 3x = 6 + 30

combine like terms: -30 + 30 = 0

0 + 3x = 6 + 30

3x = 6 + 30

combine like terms: 6 + 30 = 36

3x = 36

divide each side by '3'.

x = 12

simplifying

x = 12

this should .

answer: (3,11) is your answer.