Taylor polynomial question

Now take the first n+1 terms of the sum on the right-hand side to get the nth Taylor polynomial for ln(1-x).In summary, the Taylor Polynomial of degree n of the function f(x) at x=0 is given by Pn(x) = -1x - (x^2)/(2!) - (2(x^3))/(3!) - (6(x^4))/(4!) - (24(x^5))/(5!), or, in another form, -(x^n)/(n). This can be found by either taking the first n+1 terms of the sum of x^n
  • #1
smk037
68
1

Homework Statement



Write down the Taylor Polynomial of degree n of the function f(x) at x=0

Homework Equations



f(x) = ln(1-x)


The Attempt at a Solution



f(x) = ln(1-x)

f'(x) = (-1)((1-x)^(-1))

f``(x) = (-1)((1-x)^(-2))

f```(x) = (-2)((1-x)^(-3))

f````(x) = (-6)((1-x)^(-4))

f`````(x) = (-24)((1-x)^(-5))

Pn(x) = -1x - ((x^2)/(2!)) - ((2(x^3))/(3!)) - ((6(x^4))/(4!)) - ((24(x^5))/(5!))

now, I rechecked all my derivatives, but I still can't find a pattern to make an nth term with.

any help would be appreciated.

thanks.
 
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  • #2
1, 2, 6, 24 etc is 1!, 2!, 3!, 4! etc. So you have (n-1)!/n!. What's that?
 
  • #3
I think I got it

I got the numbers in the factorials to cancel out, and came up with

(-(x^n))/(n)
 
  • #4
Thanks a lot, Dick, I appreciate it.
 
  • #5
Thanks a lot.
 
  • #6
Another way to do it:
You know that

[tex]\sum^{\infty}_{n=0} x^n = \frac{1}{1-x}[/tex]

So by integration you get that

[tex]ln(1-x)=-\int{\sum^{\infty}_{n=0}x^n dx}[/tex]
 

1. What is a Taylor polynomial?

A Taylor polynomial is a way to approximate a function with a polynomial. It is named after the mathematician Brook Taylor and is a useful tool in calculus and other areas of mathematics.

2. How is a Taylor polynomial calculated?

A Taylor polynomial is calculated using the function's derivative at a specific point. The polynomial is then written as a sum of terms, each of which represents a derivative of the function evaluated at the given point.

3. What is the purpose of using a Taylor polynomial?

The purpose of using a Taylor polynomial is to approximate a function with a simpler polynomial. This is helpful when dealing with complex functions or when exact solutions are difficult to obtain.

4. Can a Taylor polynomial accurately represent any function?

No, a Taylor polynomial can only accurately represent a function within a certain interval around the point at which it is calculated. The accuracy of the approximation depends on the number of terms used in the polynomial.

5. How is a Taylor polynomial different from a Maclaurin polynomial?

A Taylor polynomial and a Maclaurin polynomial are both types of Taylor series, but they differ in the point at which they are evaluated. A Taylor polynomial is evaluated at a specific point, while a Maclaurin polynomial is evaluated at 0.

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