Finding the Taylor series of a function

In summary, the conversation discussed finding the Taylor series of a given function f(x) and converting it into the form of a geometric series. The attempt at a solution involved using the table method, but ended up with the same series. It was suggested to try differentiating and setting x=-1 to find the coefficients, but the result was the same. It was questioned why the professor would ask to rewrite the series in the same way if it was already a Taylor series.
  • #1
Elbobo
145
0

Homework Statement


[sorry about the formatting, I had no idea how I would latex the sigma notation]
Let f(x) = [n=1 to infinity] summation of (-1)n n2 / 3n * (x+1)n

Find the Taylor series of f(x) centered at c = -1

Homework Equations


Taylor series defined by
[n=0 to infinity] summation of f(n)(c) / n! * (x-c)n

The Attempt at a Solution


I tried converting the original function into
the form [tex]f(x) = \frac{a_{n}}{1-r}[/tex]
and using the table method to find a Taylor series but I ended up with the same series (no surprise...).

Can anyone help? I don't know how I'm supposed to convert the series into a Taylor series...
 
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  • #2
I think that is a Taylor Series. Have you tried differentiating and setting x=-1 to find the coeffients?

Mat
 
  • #3
I have, and I end up with the original series.
But if that were already a Taylor series, why would my professor waste a question to ask me to rewrite the series in the same exact way...
 

FAQ: Finding the Taylor series of a function

What is a Taylor series?

A Taylor series is an infinite sum of terms that represents a function at a specific point. It is used to approximate a function by adding together an infinite number of terms, each representing a different degree of the function's derivatives at that point.

Why do we need to find the Taylor series of a function?

The Taylor series allows us to approximate a function with polynomials, which are often easier to work with than the original function. It also helps us to better understand the behavior of a function and make predictions about its values at different points.

How do you find the Taylor series of a function?

To find the Taylor series of a function, we use a formula called the Taylor series expansion. This formula involves taking derivatives of the function at a specific point and plugging them into a specific pattern to generate the terms of the series.

What is the difference between a Taylor series and a Maclaurin series?

A Maclaurin series is a special case of a Taylor series, where the point of approximation is 0. In other words, a Maclaurin series is the Taylor series of a function at x=0. In general, a Taylor series can be centered at any point, while a Maclaurin series is always centered at 0.

Can a Taylor series represent any function?

No, not all functions can be represented by a Taylor series. Some functions have singularities or are not analytic (meaning they cannot be described by a power series). In these cases, the Taylor series would not be a valid representation of the function.

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