# Calculus II Help : Taylor/Maclaurin Series

Hey... This really sucks. I am in Calculus 2 and I have had 3 in-class exams, all 3 were A's. This last exam is take-home and it is entirely Maclaurin and Taylor series.. The only thing in the class to go over my head.

## Homework Statement

Use the function : f(x) = 1 / x^2 to answer the following questions.

#1
a. Find a formula for the sequence of values given by f^n (2). Do this by computing enough derivatives of f(x) evaluated at 2 until you see a pattern.

I got

Σ ( (-1)^(n+1) * (n+1)! ) / ( -2 * 2^(n+1) )
n=0

b. Find formula for the sequence of values given by f^n (2) / n!

I got like... ( (-1)^n (n+1)! ) / (4 * 2^n )

c. What is the Taylor Series centered at a = 2 for the function f(x) = 1/x^2 ?

Σ ( f^n*(2)*(x-2)^n ) / n!
n=0

d. What is interval of convergence for this Taylor series?

No bueno.

e. What is T4 (x) ?

f. What are T4(3) and R4 (3) ?

#2
a. Find Maclaurin Series for the function:
F(x) =
x⌠ t^2 * e^ (-t^2) dt
0⌡

*Remember : e^x =

Σ [ f^n * (a) * (x-a)^n ] / n!
n=0

I got... (something that didn't work)

[ (-1)^n * t^2 (t^(2n) ] / n!

b. Estimate value of
1⌠ x^2 * e^(-x^2) dx
0⌡
by using M9(x), the Maclaurin polynomial of degree 9.

#3

a. Find the Maclaurin series for the function f(x) = arctan ( x^3 / 3 )

b. What is the interval of convergence?

c. Find the value of the first 10 coefficient terms: c0, c1, c2, c3, c4 ... c10 for this Maclaurin series.

d. What is the value of f^21 (0), the 21st derivative evaluated at zero?

## The Attempt at a Solution

HallsofIvy
Homework Helper
Hey... This really sucks. I am in Calculus 2 and I have had 3 in-class exams, all 3 were A's. This last exam is take-home and it is entirely Maclaurin and Taylor series.. The only thing in the class to go over my head.

## Homework Statement

Use the function : f(x) = 1 / x^2 to answer the following questions.

#1
a. Find a formula for the sequence of values given by f^n (2). Do this by computing enough derivatives of f(x) evaluated at 2 until you see a pattern.

I got

Σ ( (-1)^(n+1) * (n+1)! ) / ( -2 * 2^(n+1) )
n=0
You mean the nth derivative, not f to the nth power here, don't you? f(x)= x-2 so f(2)= 1/4; f'(x)= -2x-3 so f'(2)= -1/4; f"(x)= 6x-4 so f"(2)= 3/8; f"'(x)= -12x-5 so f"'(2)= 3/16, etc. I don't see where you got that sum.

b. Find formula for the sequence of values given by f^n (2) / n!

I got like... ( (-1)^n (n+1)! ) / (4 * 2^n )
The only difference between (a) and (b) is that you have divided by n!. What happened to the sum? Why do you still have (n+1)!? (n+1)!/(n+1)= n+1.

c. What is the Taylor Series centered at a = 2 for the function f(x) = 1/x^2 ?

Σ ( f^n*(2)*(x-2)^n ) / n!
n=0
That's the formula, yes, but obviously you are expected to use your answers from (a) and (b)!

d. What is interval of convergence for this Taylor series?

No bueno.
I can think of two ways to find the radius of convergence.
a) Use the ratio test for convergence
b) What is the distance from x= 2 to the point where f(x) "blows up"?

e. What is T4 (x) ?

f. What are T4(3) and R4 (3) ?
If you were able to do (c) why not just set x= 2?

#2
a. Find Maclaurin Series for the function:
F(x) =
x⌠ t^2 * e^ (-t^2) dt
0⌡

*Remember : e^x =

Σ [ f^n * (a) * (x-a)^n ] / n!
n=0

I got... (something that didn't work)

[ (-1)^n * t^2 (t^(2n) ] / n!
First it should be a function of x, not t! Did you forget to integrate?

b. Estimate value of
1⌠ x^2 * e^(-x^2) dx
0⌡
by using M9(x), the Maclaurin polynomial of degree 9.

#3

a. Find the Maclaurin series for the function f(x) = arctan ( x^3 / 3 )

b. What is the interval of convergence?

c. Find the value of the first 10 coefficient terms: c0, c1, c2, c3, c4 ... c10 for this Maclaurin series.

d. What is the value of f^21 (0), the 21st derivative evaluated at zero?

At try 3!