Convergence of a summations, Table?

In summary, convergence of a summation refers to the behavior of a series where the terms are added together, and it is said to converge if the sum of the terms approaches a finite value as the number of terms increases. The most commonly used test for convergence of a series is the comparison test, and other tests include the ratio test, root test, and integral test. A table of convergence is a tool used to determine the convergence or divergence of a series by listing the values of the terms and their corresponding partial sums. There is a difference between absolute and conditional convergence, where absolute convergence refers to a series where the absolute value of each term is decreasing and the series converges, while conditional convergence refers to a series where the absolute value of
  • #1
perryben
8
0
Are there any good tables showing the convergence of common summations? I am convolving some discrete time signals and they can get very tricky. Thanks all
 
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  • #2
Try Gradsteyn & Rytzik which has plenty of sums and products in the first part.

Or simply use the latest version of Maple or Mathematica to see whether they can be calculated to a finite value.

Daniel.
 
  • #3


There are indeed many tables available that show the convergence of common summations. Some examples include the Handbook of Mathematical Functions by Abramowitz and Stegun, the CRC Standard Mathematical Tables and Formulae, and the NIST Digital Library of Mathematical Functions. These resources provide tables for various types of summations, including geometric, arithmetic, and hypergeometric summations. Additionally, many textbooks and online resources also offer tables for common summations.

In terms of convolving discrete time signals, there are also tables and resources available specifically for this topic. For example, the Digital Signal Processing Handbook by Madisetti and Williams includes tables for discrete convolution, as well as information on how to perform convolutions using various methods such as the fast Fourier transform (FFT) or the overlap-add method.

In addition to using tables, it may also be helpful to understand the properties of summations and their convergence. This can assist in determining the convergence of a particular summation and finding the appropriate method for calculating it. It may also be beneficial to consult with a math or signal processing expert for guidance on specific convolutions that may be particularly tricky.

Overall, there are many resources available for tables and information on the convergence of summations, including those specifically related to convolving discrete time signals. By utilizing these resources and understanding the properties of summations, you can effectively handle complex convolutions and ensure accurate results.
 

What is the definition of convergence of a summation?

Convergence of a summation refers to the behavior of a series where the terms are added together. It is said to converge if the sum of the terms approaches a finite value as the number of terms increases. If the sum of the terms does not approach a finite value, the series is said to diverge.

What is the test for convergence of a series?

The most commonly used test for convergence of a series is the comparison test, where the series is compared to another series whose convergence is already known. Other tests include the ratio test, root test, and integral test.

What is a table of convergence?

A table of convergence is a tool used to determine the convergence or divergence of a series by listing the values of the terms and their corresponding partial sums. It can help identify patterns and provide visual representation of the behavior of the series.

What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series where the absolute value of each term is decreasing and the series converges. Conditional convergence refers to a series where the absolute value of each term is decreasing, but the series itself may converge or diverge.

How can I use a table of convergence to determine the convergence of a series?

To use a table of convergence, you can observe the values of the terms and their corresponding partial sums. If the values of the partial sums are approaching a finite value, the series is said to converge. If the values are increasing or do not approach a finite value, the series is said to diverge.

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