Calculus II Help : Taylor/Maclaurin Series

[vdeity]

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.

Please help me out with these problems!

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?

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

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.

Please help me out with these problems!

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 n^th derivative, not f to the n^th 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?

Homework Equations

The Attempt at a Solution

At try 3!