5.4.2		Numerical series (convergence tests and summation)

Encouraging Mathematical Inquisitiveness, Carl L. Main, 1:1, 1970, 32-36, 5.2.2
Telescoping Sums and the Summation of Sequences, G. Baley Price, 4:2, 1973, 16-29, 6.3
Calculus by Mistake, Louise S. Grinstein, 5:4, 1974, 49-53, C, 5.1.2, 5.1.4, 5.2.2, 5.2.3, 5.2.5, 5.2.10, 5.6.1, 5.7.2
A Precalculus Unit on Area Under Curves, Samuel Goldberg, 6:4, 1975, 29-35, 0.7
An Interesting Use of Generating Functions, Aron Pinker, 6:4, 1975, 39-45, 0.6, 9.5
A Helpful Device: or One More Use for Pascal's Triangle, Robert Rosenfeld, 8:3, 1977, 188-191, C, 0.9
A Coin Game, Thomas P. Dence, 8:4, 1977, 240-246, 9.9, 9.10
Geometric Series on the Gridiron, Andris Niedra, 9:1, 1978, 18-20
A Note on Infinite Series, Louise S. Grinstein, 9:1, 1978, 46-47, C
A Note on the Integral Test, Peter A. Lindstrom, 9:2, 1978, 105-106, C
Flow Chart for Infinite Series, Thomas W. Shilgalis, 9:3, 1978, 191, C
On Sum-Guessing, Mangho Ahuja, 10:2, 1979, 95-99
The Sum of the Reciprocals of the Primes, W. G. Leavitt, 10:3, 1979, 198-199, C
Calculator-Demonstrated Math Instruction, George McCarty, 11:1, 1980, 42-48, 5.1.1, 5.2.2, 9.6
An Investment Approach to Geometric Series, Robert Donaghey and Warren Gordon, 11:2, 1980, 120-121, C
A Precalculus Approximation of n!, Norman Schaumberger, 11:3, 1980, 202-204, C, 0.2
Summation of Finite SeriesÑA Unified Approach, Shlomo Libeskind, 12:1, 1981, 41-50, 6.3
Some Sum of Sums, Gerald Lenz, 12:3, 1981, 208-209, C
The Saint Petersburg Paradox and Some Related Series, Allan J. Ceasar, 12:5, 1981, 306-308
Infinite Series Flow Chart for the Sum of a(n), Franklin Kemp, 13:3, 1982, 199, C
Taxes on Taxes, Thomas E. Eisner, 13:4, 1982, 266-269
A Simple Explicit Formula for the Bernoulli Numbers, F. Lee Cook, 13:4, 1982, 273-274, C
The Sums of Zeroes of Polynomial Derivatives, Michael W. Ecker, 13:5, 1982, 328-329, C, 0.7, 5.1.2
Closed-Form Formulas for Quasi-Geometric Series, Arthur C. Segal, 14:2, 1983, 118-122
Sequences, Series and Pascal's Triangle, Lenny K. Jones, 14:3, 1983, 253-256, C, 6.3
On Sums of Powers of Natural Numbers, Myren Krom, 14:4, 1983, 349-351, C, 9.1
Instant Hindsight!, Norman Schaumberger, 14:4, 1983, 351, C
Evaluating e^x Using Limits, Sheldon P. Gordon, 15:1, 1984, 63-65, 5.3.2

On Problems with Solutions Attainable in More Than One Way, Jean Pedersen and George Polya, 15:3, 1984, 218-228, 0.2, 0.4
An Almost Correct Series, R. A. Mureika and R. D. Small, 15:4, 1984, 334-338, 9.6
A Monte Carlo Simulation Related to the St. Petersburg Paradox, Allan J. Ceasar, 15:4, 1984, 339-342, 7.2, 9.10
Approximate Angle Trisection, David Gauld, 15:5, 1984, 420-422, 0.6
Inverse Functions, Ralph P. Boas, 16:1, 1985, 42-47, 5.2.1, 5.3.2
On Rearrangements of the Alternating Harmonic Series, Fon Brown and L.O.Cannon and Joe Elich and David G. Wright, 16:2, 1985, 135-138, C
A Discrete Look at 1 + 2 + ... + n, Loren C. Larson, 16:5, 1985, 369-382, 0.2, 0.9, 3.1, 3.2, 6.3
Cantor's Disappearing Table, Larry E. Knop, 16:5, 1985, 398-399, C
Sums of Rearranged Series, Paul Schaefer, 17:1, 1986, 66-70
How Far Can You Stick Out Your Neck?, Sydney C. K. Chu and Man-Keung Siu, 17:2, 1986, 122-132, 9.6
Counterexamples to a Comparison Test for Alternating Series, J. Richard Morris, 17:2, 1986, 165-166, C
A Case of True Interest, Soo Tang Tan, 17:3, 1986, 247-248, C, 0.8
Another Approach to a Class of Slowly Diverging Series, Norman Schaumberger, 17:5, 1986, 417, C
Computer Algebra Systems in Undergraduate Mathematics, Don Small and John Hosack and Kenneth Lane, 17:5, 1986, 423-433, 1.2, 5.1.4, 5.1.5, 5.2.2
The Bernoullis and the Harmonic Series, William Dunham, 18:1, 1987, 18-23, 2.2
Pi/4 and ln 2 Recursively, Frank Burk, 18:1, 1987, 51, C, 5.2.5
Behold! Sums of Arctan, Edward M. Harris, 18:2, 1987, 141, C
Generating Functions, William Watkins, 18:3, 1987, 195-211, 6.3, 9.3
Computing Pi, Harley Flanders, 18:3, 1987, 230-235, 5.2.3, 8.1
A Shorter, More Efficient Proof of the limit as n goes to infinity of [(n!)^(1/n)] / n = 1/e, Joseph Wiener, 18:4, 1987, 319, C
A Simple Proof of Series Convergence, A. R. Amir-Moez, 18:5, 1987, 410, C
Estimating the Sum of Alternating Series, James D. Harper, 19:2, 1988, 149-154
Subharmonic Series, Arthul C. Sogal, 20:3, 1989, 194-200, 9.5
The Power Rule and the Binomial Formula, Stephen H. Friedberg, 20:4, 1989, 322, C, 5.1.2
Evaluating the Sum of the Series Sum(k^j / M^k), Alan Gorfin, 20:4, 1989, 329-331, C
Sum the Alternating Harmonic Series, Dave P. Kraines and Vivian Y. Kraines and David A. Smith, 20:5, 1989, 433-435, C, 1.2
Using the Finite Difference Calculus to Sum Powers of Integers, Lee Zia, 22:4, 1991, 294-300, 5.2.1, 5.4.1
The Sum is 1, John H. Mathews, 22:4, 1991, 322, C
Summation by Parts, Gregory Fredricks and Roger B. Nelsen, 23:1, 1992, 39-42, C, 5.1.2, 5.4.1, 9.3
Summing Geometric Series by Holding a Tournament, Vincent P. Schielack, 23:3, 1992, 210-211, C
Six Ways to Sum a Series, Dan Kalman, 24:5, 1993, 402-421, 9.5
The Series n^m times x^n and a Pascal-like Triangle, David Neal, 25:2, 1994, 99-101
Sum of Squares via the Centroid, Sydney H. Kung, 25:2, 1994, 111, C
Approaches to the Formula for the nth Fibonacci Number, Russell Jay Hendel, 25:2, 1994, 139-142, C, 0.2, 4.5, 9.3, 9.5
FFF #76. Telescoping Series, Eleanor A. Maddock, 25:4, 1994, 309, F
FFF. Pi is approximately ln 4, Frank Burk, 25:4, 1994, 311, F
Sum of Alternating Series (proof by picture), Guanshen Ren, 26:3, 1995, 213, 0.9
Divergence of a Series (by picture), Sidney H. Kung, 26:4, 1995, 301, C
Sums of General Geometric Series (by picture), John Mason, 26:5, 1995, 381, C
FFF #106. The Derivative of the Sum Is the Sum of the Derivatives, Ed Barbeau, 27:4, 1996, 282, F
Bargaining Theory, or Zeno's Used Cars, James C. Kirby, 27:4, 1996, 285-286, C, 6.3
FFF #111. The Bouncing Ball, Daniel J. Scully, 27:5, 1996, 372-373, F
Some Sums of Some Significance, Martha E. Dasef and Steven M. Kautz, 28:1, 1997, 52-55, C
Divergence of the Harmonic Series by Rearrangement, Michael W. Ecker, 28:3, 1997, 209-210, C
Neither a Worst Convergent Series nor a Best Divergent Series Exists, J. Marshall Ash, 28:4, 1997, 296-297, C
Using Simpson's Rule to Approximate Sums of Infinite Series, Rick Kreminski, 28:5, 1997, 368-376
Can You Sum This Familiar Series? (Proof Without Words), Dennis Gittinger, 28:5, 1997, 393, C
Sum of Cubes (proof without words), Alfinio Flores, 29:1, 1998, 61, C
Who Cares if X^2 + 1 = 0 Has a Solution?, Viet Ngo and Saleem Watson, 29:2, 1998, 141-144, C, 0.7, 5.2.5, 6.2
FFF #135. Positive Series with a Negative Sum, William A. Simpson, 29:5, 1998, 407, F
A Novel Approach to Geometric Series, Michael W. Ecker, 29:5, 1998, 419-420, C
Harmonic Series, Andrew Cusumano, 30:1, 1999, 34, C
GabrielÕs Wedding Cake, Julian F. Fleron, 30:1, 1999, 35-38, 5.2.10
FFF #141.  Evaluation of a Sum, Joe Howard, 30:2, 1999, 130-131, F
Natural Logarithms via Long Division, Frank Burk, 30:4, 1999, 309-311, C
Things I Have Learned at the AP Reading, Dan Kennedy, 30:5, 1999, 346-355, 0.2, 5.1.1, 5.1.2, 5.2.1, 5.2.6, 6.1
The Series for e via Integration, Marc Chamberland, 30:5, 1999, 397, C
Summing Series via Integrals, Frank Burk, 31:3, 2000, 178-181
Sum of Infinite Series (Mathematics Without Words), Rick Mabry, 32:1, 2001, 19, C
Sum of a geometric series (Mathematics Without Words), Carlos G. Spaht and Craig M. Johnson, 32:2, 2001, 109, C
A series for ln k, James Lesko, 32:2, 2001, 119-122, C
WhatÕs Harmonic About the Harmonic Series?, David Kullman, 32:3, 2001, 201-203 , C
Convergence-Divergence of p-Series, Rasul Khan, 32:3, 2001, 206-208, C
Arctangent Sums, Louis Bragg, 32:4, 2001, 255-257, 5.3.1
Mathematics Without Words: The Sum of the Derivatives of the Terms of the Geometric Series, Roger Nelsen, 32:4, 2001, 257, C
Geometric Progressions Ð A Geometric Approach, Michael Strizhevsky and Dmitry Kreslavskiy, 32:5, 2001, 359-362, 0.6
Sum Rearrangements, Russell A. Gordon, 32:5, 2001, 377-380, C
Ln 2 (Mathematics Without Words), Norman Schaumberger, 33:1, 2002, 23, C, 5.3.2
FFF #182. New exponent laws, Carl Libis, 33:1, 2002, 38, F
A Tale of Two Series, Thomas J. Osler and Marcus Wright, 33:2, 2002, 99-106, 7.2
Designing a Calculus Mobile, Tom Farmer, 33:2, 2002, 131-136, 5.2.4
The Alternating Harmonic Series, Leonard Gillman, 33:2, 2002, 143-145, C
An Application of Condensation, Sidney Kung, 33:2, 2002, 168, C
Divergence of Series by Rearrangement, Bernard August and Thomas J. Osler, 33:3, 2002, 233-234, C
Convergence-Divergence of p-Series, Xianfu Wang, 33:4, 2002, 314-316, C
Investigating Possible Boundaries Between Convergence and Divergence, Frederick Hartmann and David Sprows, 33:5, 2002, 405-406, C, 9.5
FFF #200. A lopsided interval of convergence, Ed Barbeau, 34:1, 2003, 50, F
FFF #206. A series that converges and diverges, Doug Kuhlman, 34:2, 2003, 135, F
Column Integration and Series Representations, Thomas P. Dence and Joseph B. Dence, 34:2, 2003, 144-148, C, 5.2.5
The Calculus Mobile Simplification, A. K. Jobbings, 34:3, 2003, 223, C
Calculus, Pi, and the Machine Age, Susan Jane Colley, 34:4, 2003, 264-269, 5.2.4, 9.6
FFF #217. A Riemann sum, Holly M. Hoover, 34:4, 2003, 314, F, 9.5
An Improved Remainder Estimate for Use With the Integral Test, Roger B. Nelsen, 34:5, 2003, 397-399, C, 9.6
Improving the Convergence of NewtonÕs Series Approximation for e, Harlan J. Brothers, 35:1, 2004, 34-39, 5.3.2
FFF #218. Alternating madness, Ollie Nanyes, 35:1, 2004, 40-41, F
A Visual Approach to Geometric Series, Beata Randrianantoanina, 35:1, 2004, 43-47, C
FFF #223. Product of sums equal sum of products, Dan Kalman, 35:2, 2004, 123, F
Sums of Harmonic-Type Series, James Lesko, 35:3, 2004, 171-182, 5.3.4
Almost Alternating Harmonic Series, Curtis Feist and Ramin Naimi, 35:3, 2004, 183-191, 9.5
Finding the Sums of Harmonic Series of Even Order, Arpad Benyi, 36:1, 2005, 44-48
Alternating Series Test (Proof Without Words), Richard Hammack and David Lyons, 36:1, 2005, 72, C
A Geometric Series from Tennis, James Sandefur, 36:3, 2005, 224-226, C, 7.2
Another Proof for the p-Series Test, Yang Hansheng and Bin Lu, 36:3, 2005, 235-237, C
In Need of Analysis, Kevin Ferland, 26:5, 2005, 365, C
Series for the Square Root of 2, Sidney Kung, 36:5, 2005, 387, C
FFF #245. The harmonic series converges, Hongwei Chen, 37:1, 2006, 40-41, F
FFF #253. Hospitalization, Michail Caulfield, 37:4, 2006, 291, F
The Divergence of Balanced Harmonic-like Series, Carl V. Lutzer and James E. Marengo, 37:5, 2006, 364-369
Another Look at Some p-Series, Ethan Berkove, 37:5, 2006, 385-386, C
FFF #264. When the limit comparison test fails, Franciszek Prus-Wisniowski, 38:2, 2007, 132-133, F
On the Convergence of Some Modified p-Series, Dongling Deng, 38:3, 2007, 223-225, C
Proof Without Words: Geometric Series Formula, James Tanton, 39:2, 2008, 106, C
Squaring a Circular Segment, Russell A. Gordon, 39:3, 2008, 212-220, 0.4, 9.6
Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Lawrence Downey, Boon W. Ong, and James A. Sellers, 39:5, 2008, 391-394, C, 5.1.1, 5.2.5