Series Convergence Calculator

Script finds the convergence, sum, partial sum plot, radius and interval of convergence, of infinite series.

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