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

[Monoxdifly]

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?

 

[tkhunny]

Re: [ASK] Fracttion Addition

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? $\dfrac{1}{n\cdot(n+1)} = \dfrac{1}{n}-\dfrac{1}{n+1}$