Question

How do you do both parts?

Answer 1

Step-by-step explanation:

f(x) = 1 / (5 + x)

Divide top and bottom by 5.

f(x) = ⅕ / (1 + ⅕x)

f(x) = ⅕ / (1 − (-⅕x))

Write as a geometric series.

f(x) = ∑ₙ₌₀°° ⅕ (-⅕x)ⁿ

f(x) = ∑ₙ₌₀°° ⅕ (-1)ⁿ (⅕)ⁿ xⁿ

f(x) = ∑ₙ₌₀°° (-1)ⁿ (⅕)ⁿ⁺¹ xⁿ

Use ratio test:

lim(n→∞)│aₙ₊₁ / aₙ│< 1

lim(n→∞)│[(-1)ⁿ⁺¹ (⅕)ⁿ⁺² xⁿ⁺¹] / [(-1)ⁿ (⅕)ⁿ⁺¹ xⁿ]│< 1

lim(n→∞)│-1 (⅕) x│< 1

│x/5│< 1

│x│< 5

-5 < x < 5

If x = -5, ∑ₙ₌₀°° ⅕ (-1)ⁿ (⅕)ⁿ (-5)ⁿ = ∑ₙ₌₀°° ⅕ (1)ⁿ, which diverges.

If x = 5, ∑ₙ₌₀°° ⅕ (-1)ⁿ (⅕)ⁿ (5)ⁿ = ∑ₙ₌₀°° ⅕ (-1)ⁿ, which diverges.

The interval of convergence is therefore (-5, 5).