Summation of a sequence by parts.

In summary, the conversation is discussing a sequence and summation of the function f(n) with different cases depending on the value of x. It is clarified that the summation is a finite sum for every positive x and that the function has two different forms depending on whether x is an integer or not. The conversation ends with the understanding of this concept.
  • #1
r.a.c.
22
0
I hope can someone clarify this for me.

I have a sequence f(of n) which is like this:

fn(x) = [tex] 0-- if--x<\frac{1}{n+1}[/tex]
is = [tex] sin^2(x/pi)--if--\frac{1}{n+1}<=x<=\frac{ 1}{n}[/tex]
is = [tex]0--if--\frac{1}{n}<x[/tex]

(the - are for spaces because I don't know how to do it. Nothing is negative)

Then there is a summation (sigma) of f(of n). I don't understand how can there be such a summation for sequence by parts or how it would look like. Can anyone explain please?
 
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  • #2
Given any positive x, there exist N such that x< 1/(N+1) and M such that x> 1/M. That is, the sum is a finite sum, between N and M, for every x.

For example, if x= 1/3, then, Since 1/3< 1/(0+1), f0(1/3)= 0. Since 1/3< 1/(1+1), f1(1/3)= 0. Since 1/(2+1)= 1/3< 1/2, f2(1/3)= [itex]sin^2(1/(3\pi)[/itex]. Since 1/(3+1)< 1/3= 1/3, f3(1/3)= [itex]sin^2(1/(3\pi))[/itex]. Since 1/3< 1/4, f4(1/3)= 0 and fn(1/3)= 0 for all larger n.

The sum is [itex]f(x)= 2sin^2(1/(3\pi))[/itex].

In fact, it looks to me like, if x= 1/n for some integer n, [itex]f(x)= 2sin^2(1/(\pi x))[/itex] while if x is any other number, [itex]f(x)= sin^2(1/(\pi x))[/itex] since if x= 1/n, we have both x= 1/n and x= 1/((n-1)+1) and so sum two terms, while if x is not, we have only 1/(n+1)< x< 1/n for a single n.
 
  • #3
Hmmm, I think I get it. Thanks.
 

Related to Summation of a sequence by parts.

What is the concept of summation of a sequence by parts?

The summation of a sequence by parts is a mathematical method used to evaluate the sum of an infinite series. It involves breaking down the series into smaller parts and then evaluating each part separately before combining them to find the total sum.

What is the formula for summation of a sequence by parts?

The formula for summation of a sequence by parts is given as: ∑(an * (bn - bn+1)) = a1 * b1 + ∑(an+1 * (bn - bn+1)). Here, an and bn represent the terms of the series and n is the index of summation.

What is the purpose of using summation of a sequence by parts?

The main purpose of using summation of a sequence by parts is to evaluate the sum of an infinite series that cannot be solved using standard methods. It allows for a more manageable and systematic approach to finding the sum of a series by breaking it down into smaller parts.

What are the conditions for using summation of a sequence by parts?

The conditions for using summation of a sequence by parts include: the series must be infinite, the terms of the series must be positive and decreasing, and the series must converge. Additionally, the limit of the product of the terms and the difference between consecutive terms must exist.

What are some examples of using summation of a sequence by parts?

Some examples of using summation of a sequence by parts include finding the sum of the harmonic series, geometric series, and telescoping series. It can also be used to evaluate integrals and solve problems in physics and engineering involving infinite series.

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