Special “Series” (Part 1) Dr. Shildneck

Nested Radicals A nested radical expression is a radical in which the sum of a pattern of radicals are “nested” continually under the previous radical. An infinite nested radical is a nested radical that continues the pattern forever. Examples of infinite nested radicals: 6+ 6+ 6+ 6+… 1+ 1+ 1+ 1+… 1+ 2+ 1+ 2+… 1+ 2+ 3+ 4+…

Evaluating Infinite Nested Radicals The Procedure Set the expression equal to x Get the radical part by itself Square both sides Replace the infinite part (that looks exactly like what you started with) of the result with x Solve the resulting quadratic equation Eliminate the “impossible” solution [Example 1] 6+ 6+ 6+ 6+…

Evaluating Infinite Nested Radicals The Procedure Set the expression equal to x Get the radical part by itself Square both sides Replace the infinite part (that looks exactly like what you started with) of the result with x Solve the resulting quadratic equation Eliminate the “impossible” solution [Example 2] 1+ 1+ 1+ 1+…

Evaluating Infinite Nested Radicals The Procedure Set the expression equal to x Get the radical part by itself Square both sides Replace the infinite part (that looks exactly like what you started with) of the result with x Solve the resulting quadratic equation Eliminate the “impossible” solution [Example 3] 2+ 2+ 2+ 2+ 2+…

Continued Fractions A continued fraction is a real number that is expressed in the form: where a0, a1, a2, … and b1, b2, b3, … are integers.

Evaluating Continued Fractions The Procedure Set the expression equal to x Get the fractional part by itself Replace the infinite part (that looks exactly like what you started with that is inside the fraction) with x Solve the resulting equation Eliminate the “impossible” solution [Example 4]

Evaluating Continued Fractions The Procedure Set the expression equal to x Get the fractional part by itself Replace the infinite part (that looks exactly like what you started with that is inside the fraction) with x Solve the resulting equation Eliminate the “impossible” solution [Example 5]

Assignment Special Series Worksheet 1