Ohio Resource Center
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Content Supports - Activities and rich problems
Adding Up Reciprocals of Perfect Squares
Discipline
Mathematics
Grades
10, 11, 12
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Professional Commentary

Using the computational power of your calculator or computer, how might you convince yourself that the sum of the reciprocals of perfect squares, 1/1 + 1/4 + 1/9 + 1/16 + ... is equal to (pi)2/6? What about the sum of reciprocals of perfect fourth powers: 1/1 + 1/16 + 1/81 + 1/256 + ...? What about the sum of reciprocals of perfect cubes: 1/1 + 1/8 + 1/27 + 1/64 + …? This problem will convince students of the validity of a very surprising and nonintuitive result involving the relationship between the square of pi and the sum of the reciprocals of perfect squares. The approach to the problem uses calculators or computers in an essential way and will reinforce the fact that fractions can be represented by repeating decimals. It will show students the power of computer calculations to suggest results, as well as the limitations on that power. This mathematically rich problem was prepared by the Ohio Resource Center to accompany the Mathematics Program Models for Ohio High Schools developed by the Ohio Department of Education. (author/sw)


Common Core State Standards for Mathematics
Standards for Mathematical Practice
CCSS.Math.Practice.MP2 
Reason abstractly and quantitatively.
CCSS.Math.Practice.MP8 
Look for and express regularity in repeated reasoning.
High School - Algebra
Arithmetic with Polynomials and Rational Expressions
Rewrite rational expressions
HSA-APR.D.6 
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
HSA-APR.D.7 
(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
Ohio Mathematics Academic Content Standards (2001)
Patterns, Functions and Algebra Standard
Grade Level Indicators (Grade 12)
6.
Make arguments about mathematical properties using mathematical induction.
7.
Make mathematical arguments using the concepts of limit.
Principles and Standards for School Mathematics
Algebra Standard
Understand patterns, relations, and functions
Expectations (9–12)
generalize patterns using explicitly defined and recursively defined functions;