# Integration and Summation

Below are some techniques for these kinds of problems and some Putnam exams where the techniques have occurred. The figures in parentheses indicate roughly what percentage of the top scorers substantially solved the problem.

At the Sep. 22 practice session, a handout was available with problem statements and solutions. Stop by my office (211 Malott) if you'd like to pick up a paper copy. For copyright reasons, they can't be posted.

A beautifully put together pesentation of problems and solutions is:

The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary

written by Kedlaya, Poonen, and Vakil, ISBN 088385807X This is sold at the MAA website (bookstore section), for example.

A good online reference to problems and solutions is

by John Scholes which has almost all (!) problems and solutions. The MAA Putnam Competition Archive as well as Sep-Nov. issues of the American Mathematical Monthly available from JSTOR are also very useful.

Putnam problems often mix several techniques, so please don't view the breakdown into categories as complete.

Also keep in mind that manipulations for integrals often have an analogue for sums and vice versa.

### Substitution:

This is often just a start and must be combined with other techniques. But for many problems, starting with a ``natural'' substitution, perhaps because of how it transforms the domain, is a good thing to investigate the consequences of.
• 1985 A5) (30%)
• 1985 B5) (15%)
• 1987 B1) (80%)
• 1993 A5) (3%)
• 1994 A2) (85%)
• 1995 B2) (40%)

### Breaking Up a Domain/ Using Symmetry:

One clever version of this (often combined with substitution) involves adding two equal but different forms of the integral or sum and finding that the sum of the two is much more tractable. (e.g. 1987 B1 and 1999 A4)
• 1989 A2) (90%)
• 1989 B1) (75%)
• 1993 B3) (70%)
• 1998 B3) (25%)

### Generating Functions:

Formally manipulating and establishing an identity for a series whose coefficients are of interest can quickly establish identities (e.g. recursive) or closed form formulas for the coefficients.
• 1987 B2) (25%)
• 1989 A6) (2%)
• 1991 B4) (20%)
• 1992 B2) (90%)
• 1997 B4) (20%)
• 1999 A3) (55%)
• 1999 B3) (25%)

### Grouping/Breaking into Cases Based on Size:

This is often the summation version of breaking up the domain in an integral.
• 1988 B4) (10%)
• 1994 A1) (97%)
• 1997 B1) (90%)
• 1999 A4) (30%)

### Taylor Approximation/Taylor Series Recognition:

By keeping in mind the remainder, this can often be used to simplify convergence questions.
• 1988 A3) (30%)
• 1995 A2) (25%)
• 1997 A3) (20%)

### Recognizing a Derivative:

• 1992 A2) (80%)
• 1996 B6) (5%)
• 1967 A1) (Easier)

### Familiar Series:

• 1996 B4) (25%)
• 2000 A1) (40%)

### Riemann Sums:

• 1996 B2) (50%)

### Integration by Parts:

In addition to the computational side one learns in elementary calculus, this is a great tool for establishing theoretical results such as the convergence of integrals or properties of solutions to differential equations.
• 1989 B3) (50%)
• 2000 A4) (7%)

### Differentiating Integrals with Parameters:

• 1991 A5) (15%) (also inequalities)

### Interpretation in Linear Algebra:

• 1986 A6) (5%)

### Direct Computation:

After a preliminary step, a number of problems often come down to this.
• 1993 A1) (90%)

### Inequalities:

This comes up a lot elsewhere as well.
• 1993 B4) (0%)
• 1999 A5) (5%)

### Other:

• 1986 A3) (40%)
• 1987 A6) (10%)
• 1989 B6) (2%)
• 1990 B2) (20%)
• 2000 B3) (5%)

Modified: September 23, 2004