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May 14th, 2018, 07:00 PM  #1 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 2,001 Thanks: 132 Math Focus: Trigonometry  [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. 
May 14th, 2018, 11:08 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,898 Thanks: 1093 Math Focus: Elementary mathematics and beyond 
$$\sum_{k=1}^n\frac{2}{k(k+1)}=\frac{2n}{n+1}$$ You can prove it via induction. Last edited by greg1313; May 14th, 2018 at 11:16 PM. 
May 15th, 2018, 06:45 AM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,791 Thanks: 970 
Sum of the reciprocals of the 1st n triangular numbers...


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