# Power series expansion of a function of x

• Unassuming
R, so the series expansion of e^{-t^2} is also valid for all x in R.In summary, the conversation discusses finding a power series expansion for the given function and determining its interval of validity. The individual is unsure if they can take the integral of the function and asks for justification. They ultimately use the ratio test to show that the series converges for all real values of x. It is also noted that the series expansion of e^{x} is valid for all real values of x, so the series expansion of e^{-t^2} is also valid for all real values of x.
Unassuming

## Homework Statement

[Directions to problem]
Show that the function of x gives a power series expansion on some interval centered at the origin. Find the expansion and give its interval of validity.

$$\int_0^x e^{-t^2} dt$$

## The Attempt at a Solution

I have that, $$e^{-t^2} = \sum_0^{\infty} \frac{(-1)^n(t^2)^n}{n!}$$

I now am wondering whether I can take the integral of this series as follows,

$$\int_0^x \sum_0^{\infty} \frac{(-1)^n(t^2)^n}{n!} dt = \sum_0^{\infty} \frac{(-1)^n(x^{2n+1})}{n!(2n+1)}$$

Am I allowed to do that and if so, what is the justication?

---

I performed the ratio test on the result and the limit as n approached 0 was 0, and I therefore concluded that the series converges for all x in R.

I think you mean "prove the function has a power series" since the integration itself does not directly "give" the power series.

Yes, as long as you are inside the radius of convergence, you can integrate a power series term by term.

Also the series expansion of $$e^{x}$$ is valid for all x

## 1. What is a power series expansion of a function of x?

A power series expansion of a function of x is a way of expressing a function as an infinite polynomial, where each term in the polynomial is multiplied by a power of x. It is often used to approximate functions and make calculations easier.

## 2. How is a power series expansion of a function of x calculated?

A power series expansion is calculated by finding the coefficients for each term in the polynomial. This is typically done by taking derivatives of the function at a specific point, known as the center of expansion, and plugging those values into the formula for a power series.

## 3. What are the applications of a power series expansion of a function of x?

Power series expansions have many applications in mathematics and science. They are commonly used in physics and engineering to approximate complex functions and make calculations easier. They are also used in computer science and data analysis to model and analyze data.

## 4. What is the difference between a power series expansion and a Taylor series expansion?

A power series expansion is a type of Taylor series expansion, where the center of expansion is typically set to 0. However, a Taylor series can use any point as the center of expansion, while a power series specifically uses 0. Additionally, a Taylor series may have a finite number of terms, while a power series is infinite.

## 5. How accurate is a power series expansion of a function of x?

The accuracy of a power series expansion depends on the function being approximated and the number of terms used in the expansion. In some cases, a power series can provide an exact representation of a function, while in others, it may only provide a good approximation within a certain range of values. Generally, the more terms used in the expansion, the more accurate the approximation will be.

• Calculus and Beyond Homework Help
Replies
2
Views
165
• Calculus and Beyond Homework Help
Replies
1
Views
224
• Calculus and Beyond Homework Help
Replies
3
Views
398
• Calculus and Beyond Homework Help
Replies
2
Views
699
• Calculus and Beyond Homework Help
Replies
1
Views
187
• Calculus and Beyond Homework Help
Replies
9
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
524
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
17
Views
588
• Calculus and Beyond Homework Help
Replies
8
Views
975