Difference between revisions of "2006 GCTM State Tournament Problems/Individual Problem 46"
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== Problem == | == Problem == | ||
− | Find the exact value of the infinite series <math>\sum_{k=0}^{\infty} \ | + | Find the exact value of the infinite series <math>\sum_{k=0}^{\infty} \cot^{-1} ( k^2 + k + 1 )</math>. |
== Solution == | == Solution == |
Latest revision as of 21:51, 10 January 2023
Problem
Find the exact value of the infinite series .
Solution
Motivated by the formula for the subtraction of cotangents, we observe that
Thus
,
which is a telescoping series equal to . We note that
, so our infinite sum is equal to
, which is equal to
. Q.E.D.