15. Power Series

30. b. By differentiating the series in part (a) term by term, show that

Σ(from n=1 to ∞) n / (n + 1)! = 1.

How would you approximate e⁻⁰ᐧ⁶ using the Taylor series for eˣ?

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero. 

b. Estimate f(0.2) and give a bound on the error in the approximation.

f(x) = tan x ≈ x

Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.

f. e⁻²ˣ

A. p₂(x)= 1 + 2x + 2x²

B. p₂(x) = 1 − 6x + 24x²

C. p₂(x) = 1 + x − x²/2

D. p₂(x) = 1 − 2x + 4x²

E. p₂(x) = 1 − x + (3/2)x²

F. p₂(x) = 1 − 2x + 2x²

Use of Tech Linear and quadratic approximation

a. Find the linear approximating polynomial for the following functions centered at the given point a.

b. Find the quadratic approximating polynomial for the following functions centered at a.

c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.

f(x)=e⁻²ˣ, a=0; approximate e⁻⁰ᐧ².

Taylor series and interval of convergence

a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

b. Write the power series using summation notation.

f(x) = 1/x², a=1

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. The function f(x) = csc x has a Taylor series centered at π/2.

Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.

b. Integrate the series to find a Taylor series for Si.

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero. 

b. Estimate f(0.2) and give a bound on the error in the approximation.

f(x) =√(1+x) ≈ 1 + x/2

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The function f(x) = √x has a Taylor series centered at 0.

Taylor series and interval of convergence

a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x)=2/(1−x)³, a=0

Taylor series

b. Write the power series using summation notation.

f(x)=sin x, a = π/2

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. Let f(x)=x⁵−1 The Taylor polynomial for f of order 10 centered at 0 is f itself.

Taylor series

a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x) = 2ˣ, a = 1

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

ƒ(x) = 1/(4 + x²), a = 0