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Find the sum \(\displaystyle \frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \frac{4}{5!} + \dotsb\)

 Jul 7, 2020
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We notice that the n-th term is \(\dfrac{n}{(n + 1)!}\)

 

If we did some sort of arithmetic trick on it:

\(\dfrac{n}{(n + 1)!} = \dfrac{(n + 1) - 1}{(n + 1)!} = \dfrac{(n + 1)}{(n + 1)(n!)} - \dfrac{1}{(n + 1)!} = \dfrac1{n!} - \dfrac1{(n + 1)!}\)

 

That means, the sum is \(\displaystyle\sum_{n = 1}^\infty \left(\dfrac1{n!} - \dfrac1{(n + 1)!}\right)\).

 

To evaluate general infinite sums, we consider the partial sum first, then take the limit. 

 

So first, we consider \(\displaystyle\sum_{n = 1}^N \left(\dfrac1{n!} - \dfrac1{(n + 1)!}\right)\).

 

Listing the terms, 

\(\displaystyle\sum_{n = 1}^N \left(\dfrac1{n!} - \dfrac1{(n + 1)!}\right) = \left(\dfrac1{1!} - \dfrac1{2!}\right) + \left(\dfrac1{2!} - \dfrac1{3!}\right) + \left(\dfrac1{3!} - \dfrac1{4!}\right) + \cdots + \left(\dfrac1{N!} - \dfrac1{(N+1)!}\right)\)

 

As you may have noticed, the terms in the middle eliminates each other nicely. The only terms left are the first, and the last.

 

\(\displaystyle\sum_{n = 1}^N \left(\dfrac1{n!} - \dfrac1{(n + 1)!}\right) = \dfrac1{1!} - \dfrac1{(N + 1)!} = 1 - \dfrac1{(N + 1)!}\)

 

Next, we will take \(\displaystyle\lim_{N\to \infty}\) on both sides.

 

(If you don't know limits, don't panic. The above notation just means "N is a really large number".)

 

\(\displaystyle\lim_{N\to \infty}\displaystyle\sum_{n = 1}^N \left(\dfrac1{n!} - \dfrac1{(n + 1)!}\right)= \displaystyle\lim_{N\to \infty}\left(1 - \dfrac1{(N + 1)!}\right)\\ \displaystyle\sum_{n=1}^\infty \dfrac{n}{(n + 1)!} = 1 - \dfrac1{\text{Really big number}} = 1\)

 

The required sum is 1.

 Jul 7, 2020

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