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May 14th, 2018, 05:49 PM   #1
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[ASK] Fraction Addition

Does anyone know how to add these fractions?
$\displaystyle \frac1{1\times2}+\frac1{2\times3}+\frac1{3\times4} +…+\frac1{2009\times2010}$
I believe making them in $\displaystyle \frac12+\frac1{6}+\frac1{12}+….+\frac1{421890}$ form isn’t the correct approach.
Is there anything we can cancel out?

Last edited by Monox D. I-Fly; May 14th, 2018 at 05:52 PM.
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May 14th, 2018, 06:17 PM   #2
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May 14th, 2018, 10:01 PM   #3
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$$\sum_{k=1}^n\frac{1}{k(k+1)}=\frac{n}{n+1}$$

You can prove it via induction.

Last edited by greg1313; May 14th, 2018 at 10:16 PM.
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May 14th, 2018, 10:48 PM   #4
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Quote:
Originally Posted by greg1313 View Post
$$\sum_{k=1}^n\frac{1}{k(k+1)}=\frac{n}{n+1}$$

You can prove it via induction.
Or use telescoping series and
$$\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$$
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