Starting a Taylor series problem, .

Thread starter ArcainineFalls531
Start date Apr 25, 2006
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In summary, a Taylor series is a mathematical technique used to approximate a function with a polynomial. To start a Taylor series problem, a point must be chosen and the function's derivatives at that point must be calculated and plugged into the Taylor series formula. This formula includes terms for each derivative, with the number of terms determining the accuracy of the approximation. While a Taylor series can be used in various fields, it can only approximate the value of a function and will never be exact unless the function is a polynomial itself.

Apr 25, 2006

ArcainineFalls531

Compute the Taylor series for f(x)= sq root (x) about x=1. Determine where the series sconverges absolutely, converges conditionally, and diverges. Hint: 2(k!)=2*4*6...(2k-2)*2k. Also 1<2, 3<4, 5<6,..., 2k-1<2k should help you out with a comparision.

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Apr 25, 2006

Jameson

Gold Member

MHB

Use Taylor's theorem. [tex]f(x)=\sum_{n=0}^{\infty}\frac{f^{n}(a)*(x-a)^n}{n!}[/tex].

1. What is the purpose of a Taylor series?

A Taylor series is a mathematical technique used to approximate a function with a polynomial. It is often used in calculus to approximate complex functions and make them easier to work with.

2. How do you start a Taylor series problem?

To start a Taylor series problem, you must first choose a point (often denoted as "a") where you want to approximate the function. Then, you calculate the function's derivatives at that point and plug them into the Taylor series formula.

3. What is the formula for a Taylor series?

The formula for a Taylor series is:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
where f'(a), f''(a), f'''(a), etc. are the derivatives of the function at the chosen point "a".

4. Can a Taylor series be used to find the exact value of a function?

No, a Taylor series can only approximate the value of a function. The more terms you include in the series, the closer the approximation will be to the actual value. However, it will never be exact unless the function is a polynomial itself.

5. In what situations would you use a Taylor series?

A Taylor series is commonly used in calculus to approximate functions, especially in situations where the function is complex or difficult to work with. It can also be used in physics and engineering to approximate physical phenomena. Additionally, it can be used in computer science to develop algorithms and models.