An Alternate Approach to Alternating Sums: A Method to DIE For
Positive sums count. Alternating sums match. Alternating sums of binomial coefficients, Fibonacci numbers, and other combinatorial quantities are analyzed using sign-reversing involutions. In particular, we Describe the quantity being considered, match positive and negative terms through an Involution, and count the Exceptions to the matching rule (the method of D.I.E.). Careful use of this technique often results in nice generalizations. Any sum arising from the Principle of Inclusion-Exclusion (P.I.E.), such as the number of derangements, can be understood using D.I.E. too.
College Mathematics Journal
Benjamin, Arthur T. and Quinn, Jennifer J., "An Alternate Approach to Alternating Sums: A Method to DIE For" (2008). SIAS Faculty Publications. 213.