What is the Closed Form of a Summation of Sinusoidal Functions?

In summary, the problem is to find a closed form for the summation of sin(x) + sin(3x) + sin(5x) + ... + sin((2n+1)x). The suggested solution is sin^2(nx)/sin(x), which has been tested and seems to work. However, the person is having trouble proving it and is seeking guidance on how to use mathematical induction to do so. They are trying to show that the sum of this formula for n terms is equal to sin^2((n+1)x)/sin(x).
  • #1
Frillth
80
0

Homework Statement



I am looking for a closed form of the summation:
sin(x) + sin(3x) + sin(5x) + ... + sin((2n-1*)x)

Homework Equations



None.

The Attempt at a Solution



Through a complete stroke of luck, I believe I have arrived at the correct solution: sin^2(nx)/sin(x)
I have tested this for many different cases, and I believe it is correct. However, I am having a hard time proving that it is. Can anybody point me in the right direction?
 
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  • #2
How did you prove that?


[EDIT: The last term should be sin(2n+1)x not 2n-1
 
  • #3
Seems to me that mathematical induction would be an obvious thing to try...
 
  • #4
I'm a little rusty on my induction skills. Is this what I need to do?

1. Show that:
sin^2(nx)/sin(x) + sin((2(n+1)-1)x) = sin^2((n+1)x)/sin(x)

2. Show that my formula works for any specific case.
 
  • #5
You need to show that

[tex]\sum_{n=0} ^N sin(2n-1)x= \frac{sin^2(Nx)}{sinx}[/tex]then add the (N+1)th term to each side and show that is can be written as

[tex]\frac{sin^2((N+1)x)}{sinx}[/tex]
 
Last edited:
  • #6
I've been trying to get these two sides equal, but I'm not coming up with anything. Which identities should I be using to tackle this problem?
 

1. What is a closed form of a summation?

A closed form of a summation is an algebraic expression that provides a direct and concise way to compute the sum of a series of numbers. It is also known as an explicit formula or a closed-form expression.

2. How is a closed form of a summation different from a recursive formula?

A closed form of a summation provides the exact value of the sum, while a recursive formula requires the previous terms to calculate the next term in the series. A closed form is more efficient for computing the sum, especially for large series.

3. What are some common examples of closed forms of summations?

Some common examples of closed forms of summations include the arithmetic series formula (sum of consecutive integers), geometric series formula (sum of a geometric sequence), and the binomial series formula (sum of binomial coefficients).

4. How do you determine the closed form of a summation?

To determine the closed form of a summation, you need to find a pattern or formula for the series. This can be done by examining the terms and trying to find a relationship between them. You can also use mathematical techniques such as induction or the method of differences to find the closed form.

5. What are the benefits of using a closed form of a summation?

The benefits of using a closed form of a summation include faster and more efficient computation of the sum, especially for large series. It also allows for easy manipulation and analysis of the series, making it useful in various mathematical and scientific applications.

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