# Solving Summation of sin with n^2 - Svensl

• svensl
In summary, The conversation is about solving the equation \sum_{n=0}^{K-1}\frac{sin(2\pi n^2\Delta)}{n} and \sum_{n=0}^{K-1}\frac{e^{j n^2 x}}{n}, where x=2\pi \Delta and delta is a number between (0, 1(, for large values of K. One suggestion is to use a well-chosen function with poles at certain places in the complex plane to turn the sum into an integral and use Jordan's lemma to solve it. However, this approach has not been explored yet.

#### svensl

Hello,
Can anyone give some hints on how to solve this:

$$\sum_{n=0}^{K-1}\frac{sin(2\pi n^2\Delta)}{n}$$

It's just the n^2 that complicates things. I tried re-writing it as

$$Im\sum_{n=0}^{K-1}\frac{e^{j n^2 x}}{n}$$,

where $$x=2\pi \Delta$$
but I cannot solve this either.

Thanks,
svensl

Last edited:
What is delta? If it is an integer than sin(2*pi*k) for any integer k is equal to 0.

Delta is a number between (0, 1(.
BTW, K will later be taken to infinity if that makes a difference.

Perhaps some well choosen function which has poles at certain places in the complex plane to give that summation as residues might be useful? Then you can use a contour integral and Jordans lemma to turn that sum into an integral along the Reals somehow?

That's without putting pen to paper so I might be way off.

## 1. What is the formula for solving summation of sin with n^2 - Svensl?

The formula for solving summation of sin with n^2 - Svensl is ∑sin(n^2 - Svensl).

## 2. How do I determine the value of n in the summation?

The value of n in the summation is determined by the upper limit of the summation. For example, if the upper limit is n=10, then the summation will include values of n from 1 to 10.

## 3. Is there a specific method or algorithm for solving this type of summation?

Yes, there are several methods and algorithms that can be used to solve summations, including the summation rule, the telescoping method, and the substitution method. It is important to choose the method that best suits the specific summation being solved.

## 4. Can I use a calculator to solve this summation?

Yes, you can use a calculator to solve this summation. However, it is important to make sure that the calculator is set to the correct mode (degrees or radians) and that the input values are correct.

## 5. Are there any real-life applications for solving summation of sin with n^2 - Svensl?

Yes, summations are often used in mathematics, physics, and engineering to solve real-life problems such as calculating distance, velocity, and acceleration. This specific summation may be used in the study of periodic motion or in the analysis of wave frequencies.