You could try replacing ##\sin(x)## with it's representation as an infinite product.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.
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?
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}##
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.
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.
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.
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.