Taylor Series: Simple Homework Statement, Find 1st & 2nd Terms

In summary, the question asks for the first two terms in the Taylor series about x=0, but the function can't be differentiated at x=0, so there's a problem. By taking a few derivatives and using limits, it looks like the question is asking for the first two terms in the Taylor series about x=0, but with the derivatives computed at different points in the series.
  • #1
clandarkfire
31
0

Homework Statement


"Determine the first two non-vanishing terms in the Taylor series of [tex]\frac{1-\cos(x)}{x^2}[/tex] about x = 0 using the definition of the Taylor series (i.e. compute the derivatives of the function)."

So I know how compute the Taylor series about x=0; it involves finding f(0), f'(0), f''(0), etc. But In this particular case, it seems that f(0) and all the derivatives are undefined at x=0. This presents a problem.

I know that I can just replace cos(x) with it's Taylor series, which would make this easy, but the question specifically tells me not to..

What am I missing?
 
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  • #2
clandarkfire said:

Homework Statement


"Determine the first two non-vanishing terms in the Taylor series of [tex]\frac{1-\cos(x)}{x^2}[/tex] about x = 0 using the definition of the Taylor series (i.e. compute the derivatives of the function)."

So I know how compute the Taylor series about x=0; it involves finding f(0), f'(0), f''(0), etc. But In this particular case, it seems that f(0) and all the derivatives are undefined at x=0. This presents a problem.

I know that I can just replace cos(x) with it's Taylor series, which would make this easy, but the question specifically tells me not to..

What am I missing?

I believe the question would want you to use the Taylor expansion of cos(x). The question doesn't say not to anywhere.
 
  • #3
I don't think so; it says to compute the derivatives of the function. Also, part b of the question asks me to use the Taylor expansion of cos(x) and compare it with the result from this part.
 
  • #4
clandarkfire said:
I don't think so; it says to compute the derivatives of the function. Also, part b of the question asks me to use the Taylor expansion of cos(x) and compare it with the result from this part.

Ah that's an interesting approach then. Start taking a few derivatives and rather than considering what's happening precisely at ##x=0##, use limits to your advantage ( You'll notice a pattern by the 3rd and 5th derivatives ).
 

1. What is a Taylor Series?

A Taylor Series is a mathematical representation of a function in the form of an infinite sum of terms. It is commonly used to approximate a function at a specific point by using values of the function and its derivatives at that point.

2. How do you find the first term of a Taylor Series?

The first term of a Taylor Series is simply the value of the function at the point where the series is centered. In other words, it is the constant term in the series.

3. How do you find the second term of a Taylor Series?

The second term of a Taylor Series is found by taking the first derivative of the function at the point where the series is centered, and then multiplying it by the distance from the center point to the point of approximation. This term is also known as the linear term.

4. Why is it important to know the first and second terms of a Taylor Series?

Knowing the first and second terms of a Taylor Series allows us to approximate the value of a function at a specific point, which can be useful in many applications such as physics, engineering, and economics. It also helps us understand the behavior of a function near a particular point.

5. Can a Taylor Series accurately represent any function?

No, a Taylor Series can only accurately represent functions that are infinitely differentiable, meaning that their derivatives exist for all orders. Functions that have discontinuities or singularities cannot be accurately represented by a Taylor Series.

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