MHB [ASK] Find 1/(1×2)+1/(2×3)+1/(3×4)+…+1/(2009×2010)

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The discussion centers on finding the sum of the series 1/(1×2) + 1/(2×3) + 1/(3×4) + … + 1/(2009×2010). Participants question the effectiveness of transforming the fractions into simpler forms and explore whether any terms can be canceled. The concept of partial fraction decomposition is introduced, highlighting that 1/(n(n+1)) can be expressed as 1/n - 1/(n+1). This approach simplifies the summation process, potentially leading to a clearer solution. The conversation emphasizes the need for effective methods in summing such series.
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Does anyone know how to add these fractions?
[math]\frac1{1\times2}+\frac1{2\times3}+\frac1{3\times4}+…+\frac1{2009\times2010}[/math]
I believe making them in [math]\frac12+\frac1{6}+\frac1{12}+….+\frac1{421890}[/math] form isn’t the correct approach.
Is there anything we can cancel out?
 
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Re: [ASK] Fracttion Addition

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

What does Partial Fraction decomposition do for us? $\dfrac{1}{n\cdot(n+1)} = \dfrac{1}{n}-\dfrac{1}{n+1}$
 

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