please help and show work im having trouble understanding

Question

question 1) Determine a Taylor series around x=0 for \int (sin x )/(x)

question 2)  Find sum of geometric series \sum_(n=0)^(\infty ) (3(2)^(n+1))/(4^(n))

Answer 1

sinx = x - x³/3! + x⁵/5! - x⁷/7! + ...

So, sinx / x = 1 - x²/3! + x⁴/5! - x⁶/7! + ...

Therefore, ∫ [sinx / x]dx = x - (1/3)(x³/3!) + (1/5)(x⁵/5!) -(1/7)(x⁷/7!) + ...

= ∑_{(n=1 to ∞)} [(-1)ⁿ⁺¹(1/((2n-1)(2n-1)!)x^2n-1]

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3(2ⁿ⁺¹)/(4ⁿ) = 3(2)(2ⁿ)/(4ⁿ) = 6(1/2)ⁿ

So, we have 6∑_{(n=0 to ∞)} (1/2)ⁿ = 6[ 1 / (1 - 1/2)] = 12