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HMMT February

Algebra Difficulty 4.9 AIME Find the answer

Find the sum of the infinite series 132−12(112−132)+152−32(132−152)+172−52(152−172)+\frac{1}{3^{2}-1^{2}}\left(\frac{1}{1^{2}}-\frac{1}{3^{2}}\right)+\frac{1}{5^{2}-3^{2}}\left(\frac{1}{3^{2}}-\frac{1}{5^{2}}\right)+\frac{1}{7^{2}-5^{2}}\left(\frac{1}{5^{2}}-\frac{1}{7^{2}}\right)+

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

1−13+13−15+⋯=11-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\cdots=1.

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