How Can I Find the Power Series Representation of the Given Integral Function?

In summary, the conversation discusses finding the series representation of a given function using the power series of elementary functions. The speaker mentions using the Maclaurin series and differentiating multiple times to find a pattern, but is looking for an easier way using the power series of elementary functions. They mention the power series for ##e^x## and ##\frac{1}{1+x}##, but are unsure how to use them in this context.
  • #1
girolamo
6
0
Hi, I'm trying to find the series representation of [tex] f(x)=\int_{0}^{x} \frac{e^{t}}{1+t}dt [/tex]. I have found it ussing the Maclaurin series, differenciating multiple times and finding a pattern. But I think it must be an eassier way, using the power series of elementary functions. I know that [tex]e^{x}=\sum_{0}^{\infty}\frac{x^{n}}{n!}[/tex] and [tex]\frac{1}{1+x}=\sum_{}^{\infty}(-1)^{n}x^{n}[/tex] but I don't know how to use it here. Thanks

(Don't hesitate to correct my english)
 
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  • #2
If ##f(x) = \sum_{n=0}^\infty a_n x^n## then
##\frac{f}{1-x} = \sum_{n=0}^\infty \sum_{j=0}^n a_j x^n##.
 

Related to How Can I Find the Power Series Representation of the Given Integral Function?

1. What is a power series representation?

A power series representation is a mathematical expression that represents a function as an infinite sum of terms. Each term in the series is a polynomial with increasing powers of a variable, usually denoted by x.

2. How do you find the power series representation of a function?

To find the power series representation of a function, one can use the Taylor series expansion. This involves finding the derivative of the function at a specific point and then plugging in the values into the general formula for a power series. The resulting series is the power series representation of the function.

3. What is the radius of convergence in a power series representation?

The radius of convergence is the distance from the center point where the power series is valid and converges to the original function. It is represented by the variable R and can be calculated using the ratio test or the root test.

4. Can a function have multiple power series representations?

Yes, a function can have multiple power series representations. This is because a function can be represented as a power series by using different center points, resulting in different radii of convergence. However, all of these representations will converge to the same function within their respective radius of convergence.

5. What are some real-world applications of power series representation?

Power series representation has many practical applications in fields such as physics, engineering, and economics. It is commonly used to model and approximate complex functions, such as temperature variation, economic growth, and electrical circuits. It is also used in numerical analysis and computer algorithms to solve problems that involve continuous functions.

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