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May 14th, 2018, 07:00 PM   #1
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[ASK] Fraction Addition with Arithmetic Series as Denominators

Does anyone know how to add these fractions?
$\displaystyle \frac11+\frac1{1+2}+\frac1{1+2+3}+…+\frac1{1+2+3 +…+2009}$
Like my previous question, I believe this one also has something which can be canceled out, and the denominators contain arithmetic series. Can this series be used to make some sorts of shortcuts?

Last edited by greg1313; May 14th, 2018 at 11:49 PM.
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May 14th, 2018, 11:08 PM   #2
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$$\sum_{k=1}^n\frac{2}{k(k+1)}=\frac{2n}{n+1}$$

You can prove it via induction.
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Last edited by greg1313; May 14th, 2018 at 11:16 PM.
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May 15th, 2018, 06:45 AM   #3
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Sum of the reciprocals of the 1st n triangular numbers...
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