Power series representing ∫sinx/x

You start with the Maclaurin series for sin(x), divide everything by x, and then integrate the entire summation. In summary, to find the Power Series representing g(x)=∫sin(x)/x, you start with the Maclaurin series for sin(x), divide everything by x, and then integrate the entire summation.
  • #1
Ryantruran
9
0

Homework Statement



Find the Power Series representing
g(x)=∫sin(x)/x


Homework Equations



sin(x)= x-(x^3/3!)+(x^5/5!)-(x^7/7!)


The Attempt at a Solution



I Havent attempted yet but was wondering if you start with the maclaurin series of sin(x)
then divide everything by x then integrate the entire summation
 
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  • #2
Ryantruran said:

Homework Statement



Find the Power Series representing
g(x)=∫sin(x)/x


Homework Equations



sin(x)= x-(x^3/3!)+(x^5/5!)-(x^7/7!)


The Attempt at a Solution



I Havent attempted yet but was wondering if you start with the maclaurin series of sin(x)
then divide everything by x then integrate the entire summation
Yes, that's what you do.
 

1. What is a power series?

A power series is a mathematical representation of a function as an infinite sum of terms, each of which is a power of the variable. It is a useful tool in calculus for approximating functions and solving equations.

2. How do you represent ∫sinx/x as a power series?

∫sinx/x can be represented as the power series Σn=0 (-1)n x2n+1/(2n+1)!, which is also known as the Maclaurin series for sinx.

3. What is the significance of using a power series to represent ∫sinx/x?

Using a power series allows us to approximate the value of ∫sinx/x at any point by simply adding up a finite number of terms. This makes it easier to calculate and work with the integral in various mathematical applications.

4. What is the convergence of the power series for ∫sinx/x?

The power series for ∫sinx/x converges for all real values of x, as it is based on the convergent geometric series formula. However, it is important to note that the series may not converge to the exact value of ∫sinx/x at certain points due to the phenomenon of Gibbs oscillation.

5. How is the power series for ∫sinx/x used in real-world applications?

The power series for ∫sinx/x is used in various fields of engineering, physics, and mathematics for approximating and solving differential equations, as well as in signal processing and data analysis. It is also a fundamental tool in the study of harmonic functions and Fourier series.

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