Finding Taylor Series for Exponential Functions

In summary, the conversation discusses using power series operations to find the Taylor series at x=0 for functions that involve x and n. The first problem in the section is xex and the person is unsure of where to start since there is no n in the function. Another person suggests multiplying the series expansion of e^x by x, which is a simple solution. The confusion arises from the use of power series operations when the problem can be solved easily without it.
  • #1
mmont012
39
0
Hello,

For the exercises in my textbook the directions state:

"Use power series operations to find the Taylor series at x=0 for the functions..."

But now I'm confused; when I see "power series" I think of functions that have x somewhere in them AND there is also the presence of an n.

Here is the first problem in the section:

1. Homework Statement

xex

Where do I go from here? There isn't an n in the function at all, so the ratio/root test won't help.

If someone could start me off in the right direction I would much appreciate it! I'm just confused at where to start...

Thank you.
 
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  • #2
Do you know or can you calculate the series expansion of ##e^x##? You could multiply it by ##x##.
 
  • #3
Is that all that you do? I wanted to do that but I thought that it was more complicated... well that is super easy then. Why do they mention power series operations? This was the part that was confusing me the most.
 
  • #4
And thank you so much for helping me!
 
  • #5
mmont012 said:
Is that all that you do? I wanted to do that but I thought that it was more complicated... well that is super easy then. Why do they mention power series operations? This was the part that was confusing me the most.
Yes, it's a pretty trivial example. A better one would be with a higher power of ##x## like maybe ##x^{10}e^x##. That would be just as easy using this method compared to doing its Taylor expansion with all those product derivatives.
 

1. What is the purpose of the Taylor Series?

The Taylor Series is used to approximate a function with a polynomial, allowing for easier calculation of values and derivatives of the function at a given point.

2. How is the Taylor Series derived?

The Taylor Series is derived by expanding a function into an infinite sum of terms, using the function's derivatives evaluated at a specific point.

3. What is the importance of convergence in Taylor Series?

Convergence in Taylor Series refers to the idea that the infinite sum of terms will approach the value of the original function as more terms are added. This is important because it determines the accuracy of the approximation and whether or not it can be used to accurately represent the function.

4. How is the convergence of Taylor Series determined?

The convergence of Taylor Series can be determined by using various tests, such as the ratio test or the integral test, to analyze the behavior of the terms in the series. If these tests show that the series converges, it can be used to approximate the function.

5. What are some common applications of the Taylor Series?

The Taylor Series has many applications in mathematics and science, including in the fields of physics, engineering, and economics. It is also used in computer algorithms for approximating functions and solving differential equations.

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