## Series Convergence Calculator

Version 1.8.1 (65.1 KB) by
Script finds the convergence, sum, partial sum plot, radius and interval of convergence, of infinite series.

Updated 23 Mar 2023

This script finds the convergence or divergence of infinite series, calculates a sum, provides partial sum plot, and calculates radius and interval of convergence of power series. The tests included are: Divergence Test (nth term test), Integral Test (Maclaurin-Cauchy test), Comparison Test, Limit Comparison Test, Ratio Test (d'Alembert ratio test), Root Test (Cauchy root test), Alternating Series Test (Leibniz test), Absolute Convergence Test, p-Series Test, Geometric Series Test, Raabe's Test, Bertrand's Test, Ermakoff's Test, Cauchy's Condensation Test, and Power Series Test. The Power Series Test uses the Ratio test, the Root test, and the Cauchy-Hadamard theorem to calculate the radius and interval of convergence of power series. All the tests have partial sum graphs, except the Power Series Test. This script will help Calculus (II or III) students with the Infinite Series chapter, Differential Equations students with Series Solutions, and Real Analysis students with Advanced Convergence Tests.
There are 15 convergence tests on the primary list (mentioned above). The Absolute Convergence Test has a second list with 3 convergence tests: Absolute Convergence with Integral Test, Absolute Convergence with Comparison Test, and Absolute Convergence with Limit Comparison Test. There are 17 convergence tests in total. All the convergence tests require an infinite series expression input, the test number chosen (from 15), and the starting k, for 12 of the tests that is all that is required to run those tests. The Absolute Convergence Test has an additional input from the Absolute Convergence Test list (from 3): Absolute Convergence with Integral Test, Absolute Convergence with Comparison Test, and Absolute Convergence with Limit Comparison Test. The 2 Comparison Tests and the 2 Limit Comparison Tests have an additional 2 inputs: whether the comparison expression is convergent or divergent, and finally the comparison expression. To enter the inputs, answer the questions at the bottom of the command window, after running the script. To the left of the title is a screen shot example of the Alternating Series Test (Theorem and Alternating Series Test description commented out to fit all information).
I wrote this script because noone else had done so, and I speculated it could garner significant downloads. I have researched and tested this script with books on infinite series, internet research, and extensively with ~22 calculus books. I initially intended this script for students, but it evolved to be so powerful, accurate, simple, and robust, that professor's download it. If anyone has questions or comments about this script, this includes employment opportunities, feel free to contact me!
[1] Daniel D. Bonar, Michael J. Khoury. "Real Infinite Series." Originally published by The Mathematical Association of America, 2006, Copyright (2006) and reprint (2018) in USA held by American Mathematical Society. Providence, Rhode Island. ISBN: 9781470447823
[2] T. J. IA. Bromwich. "An Introduction to the theory of infinite series." Alpha Editions, www.alphaedis.com (2020). ISBN: 9789354038747

### Cite As

David Cazenave (2023). Series Convergence Calculator (https://www.mathworks.com/matlabcentral/fileexchange/72141-series-convergence-calculator), MATLAB Central File Exchange. Retrieved .

##### MATLAB Release Compatibility
Created with R2019a
Compatible with R2020b and later releases
##### Platform Compatibility
Windows macOS Linux
##### Categories
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Version Published Release Notes
1.8.1

Updated for R2023a. Added Cites. Changed screen shot.

1.8.0

Cleaned up some display statements in Cauchy's Condensation Test. Modified the Overview. Changed the screen shot.

1.7.9

Updated screen shot, script description, Overview, and line numbers. Resized some plot markers.

1.7.8

Expanded capability of Comparison Tests, Limit Comparison Tests, Absolute Convergence with Integral Test, and Power Series Test. Added Cauchy's Condensation Test. Changed mfile name. Fixed Ermakoff's Test description. Updated screen shot, ...

1.7.7

Deleted some display statements from both Integral Tests. Clarified Ermakoff's Test description. Changed line numbers. Corrected MATLAB Release Compatibility. Added paragraph to script Outline.

1.7.6

Image.

1.7.5

Fixed bug in the Integral, Comparison, Limit Comparison, and Absolute Convergence Tests. Expanded capability of the 2 Comparison Tests and the 2 Limit Comparison Tests. Added Ermakoff's Test. Updated screen shot and Overview.

1.7.4

Updated and expanded the capability of the Power Series Test. Deleted tested and unneeded x2 code from Power Series Test.

1.7.3

Updated script description. Clarified some display statements.

1.7.2

Expanded capability of Bertrand's Test. Clarified some display statements.

1.7.1

Expanded capability of Bertrand's Test. Increased for loop iterations of p-Series Test.

1.7.0

Fixed bug in Bertrand's Test. Brightened up plot markers in p-Series Test and Bertrand's Test. Clarified some display statements.

1.6.9

Expanded capability of the Absolute Convergence with Integral Test, and Bertrand's Test.

1.6.8

Changed Overview.

1.6.7

Expanded capability of Raabe's Test. Fixed bug in Alternating Series Test.

1.6.6

Changed display statements in Integral, Comparison, Limit Comparison, and Absolute Convergence Tests. Added to Bertrand's Test description. Changed description at beginning of code.

1.6.5

Added more instructions to the script. Changed display statements in all of the comparison tests. Added to Bertrand's Test description.

1.6.4

Image

1.6.3

Expanded capability of Raabe's Test. Fixed bug and expanded capability of Bertrand's Test. Added to Bertrand's Test description.

1.6.2

Updated Power Series Test. Added Real Analysis to script description, Overview, and a tag. Updated line numbers.

1.6.1

Expanded capability of Integral Tests, Comparison Tests, and Limit Comparison Tests. Updated screen shot. Added Bertrand's Test tag. Added Raabe's Test tag. Changed title.

1.6.0

Expanded capability of Integral, Comparison, and Limit Comparison Tests. Fixed bug in plots, and a few more. Improved some marker colors. Enabled (-1) to a linear exponent of k. Added Bertrand's Test. Improved some mathematician bio's, ......

1.5.9

Improved robustness of the 2 Comparison Tests and fixed bug in Raabe's Test.

1.5.8

Corrected a couple of typing errors.

1.5.7

Updated the Power Series Test for R2020b.

1.5.6

Updated the Absolute Convergence Test for R2020b.

1.5.5

Updated the Limit Comparison Test, Ratio Test, Root Test, Absolute Convergence Test with the Integral Test, Absolute Convergence Test with the Limit Comparison Test, and Raabe's Test for R2020b.

1.5.4

Strengthened filters in the Absolute Convergence Test with the Integral Test, and the Alternating Series Test. Changed some display statements in the Limit Comparison Tests.

1.5.3

Fixed bug in the Absolute Convergence Test with the Integral Test. Fixed bug in the Integral Test. Reorganized display statement blocks in the 2 Comparison Tests.

1.5.2

Strengthened a filter in the Absolute Convergence Test with the Integral Test. Clarified a display statement block in the 4 comparison tests.

1.5.1

Expanded logarithmic capability of both Integral Tests, both Comparison Tests, and both Limit Comparison Tests. Expanded capability of the p-Series Test and the Geometric Series Test. Improved robustness of the Power Series Test.

1.5.0

Improved trigonometric capability of the Absolute Convergence Test with the Integral Test, and the Alternating Series Test. Adjusted Limit Comparison Test plot marker color. Fixed bug in Divergence Test, and fixed bug in Alternating Series Test.

1.0.2

Modified screenshot.

1.0.1