I cant quite get this binomial theorem question. Could someone help me please? Thank you!

Question

1a. Use the binomial theorem to compute the first four nonzero terms for the Maclaurin series for

b. Compute the interval of convergence for your series in part a.

c. Use your answer to compute

Answer 1

a. Binomial expansion of or calculating series coefficients by

gives the power series c₀ + c₁x + c₂x² +c₃x³ + ⋅⋅⋅ + c_nxⁿ where c₀ = f(0) = 2 and

for n > 0

The first 4 terms of the series are 2 + (1/12)x – (1/288)x² + (5/20736)x³

b. corrected (see comments below)

Interval of convergence

|a_n+1/a_n| = [|4 – 3(n + 1)|/(3(2³)(n + 1))]|x| = [|3n – 1|/(24n + 24)]|x|

The series only converges if |a_n+1/a_n| = ⅛|x| < 1

This means |x| < 8 and

the interval of convergence is (–8, 8)

c. (f(x) – 2)/3x = ⅓(c₁ + c₂x +c₃x² + ⋅⋅⋅ + c_nx^n–1)

The first three terms of the series are

1/36 – (1/864)x + (5/62208)x²

Answer 2

a) f(x) = (1 + x)^1/3 = 1 + 1/3x + 1/3(1/3 - 1)/2!x² + 1/3(1/3 - 1)(1/3 - 2)/3!x³ + . . . = 1 + 1/3x - 1/9x² + 5/81x³ - .

b) This series converges absolutely for IxI ≤ 1 because 1/3 > 0

c)lim x→ ∞ [(1 + x)^1/3 - 2]/(3x) = lim x→∞ [1/3(1 + x)^-2/3/3] = 0