How to calculate the series ##\sum_{x = 1}^{\infty} \frac{sin(x)}{x}##

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In summary: You can also use the non-negative, monotonic, decreasing function ## \frac{1}{n+1}## and the integral test.In summary, the conversation focused on finding a way to calculate the integral ##\int_{0}^{\infty} \frac{sin(x)}{x}## and whether the series Ʃ(1 to infinity) sinx / x converges or diverges. The suggested approaches included using a parametric factor, replacing ##\sin(x)## with its infinite product representation, expressing sine in terms of complex exponentials, using the ratio and root tests, and normalizing the function. The integral test was also mentioned as a possible method.
  • #1
shrub_broom
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Homework Statement
I have known how to calculate the integral##\int_{0}^{\infty} \frac{sin(x)}{x}##, however I find it hard to calculate this one since the similar technique(by parametric integral) cannot be simply applied on series.
Any approach or hint is welcomed.
Relevant Equations
##\sum_{x = 1}^{\infty} \frac{sin(x)}{x} = \frac{\pi - 1}{2}##
Maybe introduce a parametric factor can be help.
 
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  • #2
Is this homework ? Some activity on your part is required in PF guidelines
 
  • #3
No, just for fun.
 
  • #4
shrub_broom said:
Problem Statement: I have known how to calculate the integral##\int_{0}^{\infty} \frac{sin(x)}{x}##, however I find it hard to calculate this one since the similar technique(by parametric integral) cannot be simply applied on series.
Any approach or hint is welcomed.
Relevant Equations: ##\sum_{x = 1}^{\infty} \frac{sin(x)}{x} = \frac{\pi - 1}{2}##

Maybe introduce a parametric factor can be help.
You could try replacing ##\sin(x)## with it's representation as an infinite product.
 
  • #5
Express the sine in terms of complex exponentials.
 
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  • #6
Similar thread:

fakecop said:

Homework Statement


Determine whether the series Ʃ(1 to infinity) sinx / x converges or diverges.

Homework Equations


This question appears in the integral test section, but as far as i know the integral test can only be used for decreasing functions, right?

The Attempt at a Solution


Using the ratio test, limx->infinity sin(x+1)/(x+1)*(sinx/x)=limx->infinity x*sin(x+1)/((x+1)(sinx))
This is where i got stuck-this limit oscillates between positive infinity and negative infinity.
Using the root test, i need to find the limit of (sinx)^(1/x) as x approaches infinity, which also gets me nowhere.

we have not done taylor series yet so I'm sure there is a relatively simple approach to this question...please help?
 
  • #7
Your integral is egregiously improper. You must normalize it by redefining the function ## sinc(x)=1## when ## x=0## and ##sinc(x)= \frac{sin(x)}{x}## when ## x\neq 0##. Please see wikipedia page on sinc function.
 
  • #8
BvU said:
Similar thread:
Yes, but that is not looking for the value. @vela's method works.
 
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Related to How to calculate the series ##\sum_{x = 1}^{\infty} \frac{sin(x)}{x}##

1. What is the formula for calculating the series ##\sum_{x = 1}^{\infty} \frac{sin(x)}{x}##?

The formula for calculating this series is ##\sum_{x = 1}^{\infty} \frac{sin(x)}{x} = \frac{1}{1} + \frac{sin(2)}{2} + \frac{sin(3)}{3} + ... + \frac{sin(n)}{n}##

2. How do you determine the convergence of this series?

The series ##\sum_{x = 1}^{\infty} \frac{sin(x)}{x}## is a convergent series. This can be determined by using the alternating series test, which states that if a series alternates between positive and negative terms and the absolute value of each term decreases as x increases, then the series is convergent.

3. Can this series be simplified?

Yes, this series can be simplified by using the trigonometric identity ##sin(x) = \frac{e^{ix} - e^{-ix}}{2i}##. This results in the series ##\sum_{x = 1}^{\infty} \frac{e^{ix} - e^{-ix}}{2ix^2}##, which can be further simplified using the properties of exponents.

4. What is the relationship between the value of x and the value of the series?

As x increases, the value of the series ##\sum_{x = 1}^{\infty} \frac{sin(x)}{x}## approaches the value of ##\frac{\pi}{2}##. However, the series does not converge to this value, as it oscillates between positive and negative values as x increases.

5. How is this series used in real-world applications?

This series has many applications in mathematics and physics. It is used in Fourier analysis, which is a mathematical tool used to decompose a complex signal into simpler components. It is also used in the study of waves and vibrations, as well as in the calculation of integrals in physics and engineering.

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