Step-by-step explanation:
f(x) = (4 + x) / (1 − x)
f(x) = (5 − (1 − x)) / (1 − x)
f(x) = 5 / (1 − x) − 1
Since ∑ₙ₌₀°° (xⁿ) = 1 / (1 − x), we can write f(x) as:
f(x) = 5 ∑ₙ₌₀°° (xⁿ) − 1
f(x) = 5 (1 + ∑ₙ₌₁°° (xⁿ)) − 1
f(x) = 5 + 5 ∑ₙ₌₁°° (xⁿ) − 1
f(x) = 4 + ∑ₙ₌₁°° 5(xⁿ)
Using ratio test:
lim(n→∞)│aₙ₊₁ / aₙ│< 1
lim(n→∞)│5(xⁿ⁺¹) / 5(xⁿ)│< 1
lim(n→∞)│x│< 1
│x│< 1
-1 < x < 1
