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# Help

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I got this proplem from a friend and can't solve it. Help would be apperciated

We have a sequence where an= 1/n(n+1). (a1=1/2, a2=1/6, ...) What is the sum of the sequence: a1+a2+a3...?

Yes I know I can't write in Latex I'm sorry

Oct 23, 2023

#1
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Assuming that you are referring to the sum of the infinite sequence of a,

Let's try to expand the terms

\begin{align} a_1 &=( \frac{1}{2}) \\\\ a_2 &= \frac{1}{6} = \frac{3}{6}-\frac{2}{6} \\ &=( \frac{1}{2}) -( \frac{1}{3})\\\\ a_3 &= \frac{1}{12} = \frac{4}{12}-\frac{3}{12} \\ &=( \frac{1}{3}) -( \frac{1}{4}) \\\text{and so on...} \end{align}

Hence, we can conclude that every term of an (apart from a1equals to $$\frac {1}{n} - \frac {1}{n+1}$$

Therefore, the sum of such an infinite sequence is presented as ...$$a_1 + a_2 + a_3 + \ldots =\frac{1}{2} + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \left(\frac{1}{4} - \frac{1}{5}\right) + \ldots +\left(\frac{1}{n-2} - \frac{1}{n-1}\right) + \left(\frac{1}{n-1} - \frac{1}{n}\right))$$

... where n in this case would be infinity,

equals to...

$$a_1 + a_2 + a_3 + \ldots =\frac{1}{2} + \frac{1}{2} - \frac{1}{n} = 1 - \frac{1}{n}$$

the sum of the sequence approaches 1

further explanation (to the best of my abilities below)

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As you can imagine, even though $$\frac {1}{n}$$ never reaches 0 but continues to decrease in value,

there is a point where the value of $$\frac {1}{n}$$ would be so small

(think that when n = one trillion, $$\frac {1}{n}$$ would result in 0.000000000001

1 - 0.000000000001 = 0.99999999999

- and n, as infinity, would not stop at one trillion,

meaning that as the sequence goes on, smaller and smaller values of  $$\frac {1}{n}$$ would be subtracted from 1, )

Hence, we completely ignore the value of n, instead we can just say that

the sum of the sequence approaches 1

Oct 25, 2023
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Nice work Charlie

Melody  Oct 30, 2023
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Using an online  Sigma calculator

$$\sum_{1}^{∞}$$1/(n(n+1)    =   1

Oct 25, 2023
#3
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Found a solution: 1/n(n+1)=(1/n)-(1/n+1)

So It's (1-1/2)+(1/2-1/3)...

Everything cancels out to one

Oct 28, 2023