How do you do these questions?

-x¹⁴ / 5040

-½ < x < ½

Step-by-step explanation:

f(x) = e^(-x²)

The Taylor series for eˣ centered at 0 is:

eˣ = ∑ (1/n!) xⁿ

Substitute -x²:

e^(-x²) = ∑ (1/n!) (-x²)ⁿ

e^(-x²) = ∑ (1/n!) (-1)ⁿ x²ⁿ

The 14th degree term occurs at n=7.

(1/7!) (-1)⁷ x¹⁴

-x¹⁴ / 5040

ln(1 + x) = ∑ₙ₌₁°° (-1)ⁿ⁺¹ xⁿ / n

If we substitute 4x²:

ln(1 + 4x²) = ∑ₙ₌₁°° (-1)ⁿ⁺¹ (4x²)ⁿ / n

Using ratio test:

lim(n→∞)│aₙ₊₁ / aₙ│< 1

lim(n→∞)│[(-1)ⁿ⁺² (4x²)ⁿ⁺¹ / (n+1)] / [(-1)ⁿ⁺¹ (4x²)ⁿ / n]│< 1

lim(n→∞)│-1 (4x²) n / (n+1)│< 1

4x² < 1

x² < ¼

-½ < x < ½