• MHB
• Monoxdifly
In summary, the conversation is about adding fractions with denominators that contain arithmetic series. The given series can be simplified using the formula for the sum of an arithmetic series and can be written as S=2\left(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\cdots+\frac{1}{2009\cdot2010}\right).
Monoxdifly
MHB
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?

Monoxdifly said:
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?

We know:

$$\displaystyle \sum_{k=1}^n(k)=\frac{n(n+1)}{2}$$

And so the given series becomes:

$$\displaystyle S=2\left(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\cdots+\frac{1}{2009\cdot2010}\right)$$

## What is the purpose of this series?

The purpose of this series is to find the sum of a sequence of fractions with a specific pattern.

## What is the pattern in this series?

The pattern in this series is that each fraction in the sequence has a denominator that is the sum of the previous numbers in the sequence plus one.

## What is the value of the first term in the series?

The value of the first term in the series is 1.

## What is the value of the last term in the series?

The value of the last term in the series is 1/2009.

## What is the overall sum of the series?

The overall sum of the series is approximately 7.399.

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