Taylor Series for f(x) with nth Derivatives and Coefficients | Homework Help

[nns91]

Homework Statement

Let f be a function with derivatives of all orders and for which f(2)=7. When n is odd, the nth derivative of f at x=2 is 0. When n is even and n=>2, the nth derivative of f at x=2 is given by f⁽ⁿ⁾ (2)= (n-1)!/3ⁿ

a. Write the sixth-degree Taylor polynomial for f about x=2.
b. In the Taylor series for f about x=2, what is the coefficient of (x-2)⁽²ⁿ⁾ for n =>1 ?
c. Find the interval of convergence of the Taylor series for f about x=2. Show the work that leads to your answer

Homework Equations

Taylor series

The Attempt at a Solution

a. I got the sixth-degree series.
b. Will it just be (n-1)!/3ⁿ / (2n)! ??
c. So will the Taylor series be \sum (n-1)/3ⁿ * (x-2)^2n from n=1 to infinity ??

 

[e(ho0n3]

b. Will it just be (n-1)!/3ⁿ / (2n)! ??

No. Try again.

c. So will the Taylor series be \sum (n-1)/3ⁿ * (x-2)^2n from n=1 to infinity ??

No. See part b.

 

[nns91]

Can you give me some hints ?

 

[e(ho0n3]

You made a very simple mistake with the n's. Looking at some examples should reveal them. Try 2n = 2 and 2n = 4.

 

[nns91]

For 2n=2, the coefficient will be (n-1)!/6 right ?

So should it be (n-1)!/3^n /n! ??

 

[e(ho0n3]

For 2n=2, the coefficient will be (n-1)!/6 right ?

No. Try again and please show your work.

 

[nns91]

So for n> or =1, 2n is always even so for the coeffecient we have to use the formula that is given: (n-1)!/3^n but also the denominator contains an even factorial so it will be 2n

so will the coefficient be (n-1)!/3^n / 2n!

 

[e(ho0n3]

Still wrong. If n is even, the coefficient of the (x-2)ⁿ in the Taylor series is f⁽ⁿ⁾(2)/n! = (n-1)!/(3ⁿn!), so when you substitute n for 2k (I will use k instead of n because I believe this is what is confusing you), what do you get?

 

[nns91]

So the coefficient will be (n-1)!/(3^n*n!) ??

 

[Billy Bob]

You keep answering the question (b) that could have been asked, instead of the question (b) that actually was asked.

The question that could have been asked is this: what is the coefficient of (x-2)^n when n is even?

By the way, what is 99!/100! ?