- #1

- 6

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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?