# Taylor Series for cos(x^5) | Computing f^(90)(0) | Homework Solution

• eifphysics
In summary: This way, you can see which terms will contribute to f^(90)(0).In summary, the problem asks to find the 90th derivative of f(x)=cos(x^5) at x=0 using the Taylor series expansion. The correct approach is to substitute x^5 into the Taylor series for cos(x), keeping the same coefficients. This will give the expanded form of the series, allowing for the identification of the terms that contribute to f^(90)(0).

## Homework Statement

Let f(x)=cos(x^5). By considering the Taylor series for f around 0, compute f^(90)(0).
by the way, I don't know how super/sub script works?

## The Attempt at a Solution

I tried to substitute x^5 into x's Tyler Series form and solve for f^(90)(0), but it gave me a wrong answer.

Minjie said:

## Homework Statement

Let f(x)=cos(x^5). By considering the Taylor series for f around 0, compute f^(90)(0).
by the way, I don't know how super/sub script works?

## The Attempt at a Solution

I tried to substitute x^5 into x's Taylor Series form and solve for f^(90)(0), but it gave me a wrong answer.
Show us what you got when you did the substitution. That's the right approach.

BTW, it's Taylor series, not Tyler series.

cosx=∑(-1)^n/(2n)!*x^2n
I am not sure what the an part: (-1)^n/(2n)! will becomes when I substitute x5 into the series.

Minjie said:
cosx=∑(-1)^n/(2n)!*x^2n
I am not sure what the an part: (-1)^n/(2n)! will becomes when I substitute x5 into the series.
Nothing changes in that part. In your formula above, replace x by x5. That will be your Taylor series for cos(x5).

It might be helpful to write the new series in expanded form rather than in closed form (as a summation).

## 1. What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms, where each term is a derivative of the function evaluated at a specific point.

## 2. Why are Taylor series important?

Taylor series allow us to approximate complex functions with simpler ones, making it easier to analyze and solve problems in various fields such as physics, engineering, and economics.

## 3. What is the purpose of using a Taylor series in problem-solving?

The main purpose of using a Taylor series is to approximate a function at a specific point, which can be useful in solving problems where the function is difficult to evaluate directly.

## 4. How do you find the coefficients of a Taylor series?

The coefficients of a Taylor series can be found by taking successive derivatives of the function at the point of expansion and evaluating them at that point.

## 5. Can a Taylor series represent any function?

No, a Taylor series can only represent analytic functions, which are functions that can be expressed as a power series. Functions with discontinuities or vertical asymptotes cannot be represented by a Taylor series.