Numerical Analysis: Taylor Polynomials, Error, Bounds

In summary, the conversation discusses finding a Taylor polynomial for a general (3n)th degree and calculating the error for a 6th order Taylor polynomial approximation. The equation 1/(1-x)=1+x+x^2+x^3+...+[x^(n+1)]/(1-x) is used to find the polynomial, and the error is calculated using the difference between the function and the polynomial, with the maximum of the 7th derivative of the function on the interval [0, 0.1] as the bound for the error.
  • #1
Goomba
11
0
(a) I found the answer to be:
1/(1-x) = 1 + x + x^2 + x^3 + ... + [x^(n+1)]/(1-x) for x != 1

*Note: "^" precedes a superscript, "!=" means "does not equal"

(b) Use part (a) to find a Taylor polynomial of a general (3n)th degree for:
f(x) = (1/x)*Integral[(1/(1 + t^3), t, 0, x]

*Note: Integral[f(t), variable, lower bound, upper bound]

(c) What is the error for the approximation you found in part (b)?

(d) Find a bound for the error when a 6th order Taylor polynomial is used to approximate f(0.1).

====================================

For (b), I used (a)'s equation to find that 1/[1 - (-t^3)] = 1 - t^3 + t^6 - t^9 + t^12 - ... + (-t)^(3n) + (-t)^[3(n + 1)]
Then I integrated with respect to t with lower and upper bounds of 0 and x, respectively. I ended up with:
f(x) = 1 - (x^3)/4 + (x^6)/7 - (x^9)/10 + ... + Integral[(-t)^(3n), t, 0, x] + Integral[(-t)^[(3(n+1)]/(1+t^3, t, 0, x]

I'm confused about finding the (3n)th polynomial...

For (c), I said that the error is R_(3n)(x) = f(x) - p_(3n)(x)... But I need to find the (3n)th polynomial to finish this one...

For (d), I let n = 6, because it is the 6th order Taylor polynomial (??) to be p_6(x) = 1 - (x^3)/4 + ... The professor said to use n = 2, but I am not sure why...
 
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  • #2
The error is R_6(x) = f(x) - p_6(x). Then, the bound for the error is |R_6(0.1)| <= M*|0.1|^7, where M is the maximum of the |f^(7)(x)| on the interval [0, 0.1].
 

What is a Taylor Polynomial?

A Taylor polynomial is a mathematical approximation of a function using a finite number of terms from its Taylor series. It is used to estimate the value of a function at a specific point by using known derivatives of the function at that point.

What is the error in a Taylor Polynomial?

The error in a Taylor Polynomial is the difference between the actual value of the function and the value estimated by the polynomial. It is also known as the remainder term and is typically represented by the symbol "R".

How do you calculate the error in a Taylor Polynomial?

The error in a Taylor Polynomial can be calculated using the Lagrange form of the remainder term, which takes into account the highest order derivative of the function and the distance between the point of approximation and the center of the Taylor series.

What are error bounds in Taylor Polynomials?

Error bounds in Taylor Polynomials refer to the upper and lower limits within which the actual value of the function is guaranteed to fall, given a specific degree of the polynomial and a specific distance from the center of the Taylor series.

How are Taylor Polynomials used in Numerical Analysis?

Taylor Polynomials are used in Numerical Analysis to approximate the values of functions that are difficult or impossible to calculate exactly. They can also be used to speed up the computation of functions by using a finite number of terms instead of an infinite series.

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