Energy of a sequence - absolutely summable sequence or square summable sequence

In summary, the energy of the sequence [sin (pi/4 n)] / [(pi) (n)] is infinite and to determine if a sequence is absolutely summable or square-summable, you need to check the energy of the sequence.
  • #1
caramello
14
0
Hi,

I was doing some practise on the inverse DTFT and already derive the sequence. But then the question asks about the energy of the sequence. Let's say the sequence that I got is x(n) = [sin (pi/4 n)] / [(pi) (n)]. To find the energy, I already found something like 2 * sum from n=1 to infinity of [sin2(pi/4 n)] / [pi2 n2] but then after that I was stuck, and I don't know how to solve the summation. Can someone help me with this?

Also, how do we know if a sequence is absolutely summable or if it is square-summable from the energy?

Thank you so much!
 
Physics news on Phys.org
  • #2
The energy of the sequence x(n) = [sin (pi/4 n)] / [(pi) (n)] can be calculated as follows: Energy of x(n) = ∑|x(n)|2 = ∑[sin(π/4n) / (πn)]2 = 1/π2 * ∑[sin2(π/4n)]/(n2) = 1/π2 ∑[1 - cos(π/2n)]/(n2) = 1/π2 * ∑1/n2 - 1/π2 * ∑cos(π/2n)/n2 = 1/π2 * [∞ - 0] - 1/π2 * [0] = ∞Therefore, the energy of the given sequence is infinite. To know whether a sequence is absolutely summable or square-summable, you need to check if the energy of the sequence is finite or infinite. If the energy of the sequence is finite, then it is absolutely summable and if it is infinite, then it is square-summable.
 
  • #3


Hi,

The energy of a sequence is a measure of its power or magnitude. It can be calculated by taking the sum of the squared values of the sequence. In your case, the energy of your sequence x(n) would be given by the integral of [sin2(pi/4 n)] / [pi2 n2] from -∞ to ∞. This integral can be evaluated using techniques from calculus, such as integration by parts or the residue theorem.

Regarding your question about absolute summability and square summability, these are different concepts related to the convergence of a sequence. A sequence is said to be absolutely summable if the sum of the absolute values of its terms is finite. In other words, the sequence converges in the absolute sense. On the other hand, a sequence is said to be square summable if the sum of the squares of its terms is finite. This implies that the sequence converges in the mean-square sense.

In general, the energy of a sequence can provide information about its summability. For example, a sequence with finite energy must be absolutely summable, but the converse is not always true. A sequence with finite energy can also be square summable, but again, the converse is not always true. In your case, the energy of your sequence x(n) is infinite, so it is neither absolutely summable nor square summable.

I hope this helps clarify the concept of energy and its relationship to summability. Keep up the good work in your studies!
 

1. What is the difference between absolutely summable sequences and square summable sequences?

Absolutely summable sequences are sequences in which the sum of the absolute values of the terms converges to a finite value. Square summable sequences are sequences in which the sum of the squares of the terms converges to a finite value. In other words, absolutely summable sequences have a finite sum, while square summable sequences have a finite sum of the squares of the terms.

2. How do you determine if a sequence is absolutely summable or square summable?

A sequence is absolutely summable if the sum of the absolute values of the terms converges. This can be determined using the Cauchy criterion or the comparison test. A sequence is square summable if the sum of the squares of the terms converges. This can be determined using the Cauchy-Schwarz inequality or the comparison test.

3. What is the significance of absolutely summable and square summable sequences?

Absolutely summable and square summable sequences are important in the study of mathematical series and their convergence. They are used in various areas of mathematics, such as analysis, functional analysis, and harmonic analysis. They also have applications in physics, engineering, and other sciences.

4. Can a sequence be both absolutely summable and square summable?

Yes, a sequence can be both absolutely summable and square summable. In fact, every square summable sequence is also absolutely summable, but the converse is not necessarily true.

5. What are some examples of absolutely summable and square summable sequences?

An example of an absolutely summable sequence is the geometric series 1, 1/2, 1/4, 1/8, ... which has a finite sum of 2. An example of a square summable sequence is the sequence 1, 1/2, 1/3, 1/4, ... which has a finite sum of π²/6. Other examples include sequences of rational numbers and sequences of trigonometric functions.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
92
Replies
2
Views
337
Replies
3
Views
2K
  • Calculus
Replies
7
Views
2K
  • Calculus
Replies
1
Views
1K
  • Math Proof Training and Practice
Replies
8
Views
951
  • General Math
Replies
2
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
1
Views
1K
Back
Top