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So I was playing around with some equations and I got to this formula:

$$ \prod\limits_{n=1}^{x}\left(1+\frac{1}{n}\right)=x+1 $$

I checked it with particular values and it holds true when x is a positive integer.

I never saw this formula before. Is it just a useless formula? If not, does it have a name?

Blue
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Robert Puscasu
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    Not holding true for $x=1,2,3...$. Perhaps it should be $x+1$ on the RHS. Not very difficult to see why: $1+1/n=(n+1)/n$ – Shubham Johri Feb 19 '21 at 05:05
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    Oh, yeah, my bad. Youre right, its x+1 – Robert Puscasu Feb 19 '21 at 05:05
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    It's an example of a telescoping product (scroll down a bit to see the bit about telescoping product). Not sure about any particular name, though. – Kyan Cheung Feb 19 '21 at 05:07
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    No name for this formula, it's just the textbook example of a telescoping product – Some Guy Feb 19 '21 at 05:16
  • Telescoping sums where $b_k = a_{k+1}-a_k$ so $b_0 +b_1 + b_2 + ..... + b_n = (-a_0 + a_1) + (-a_1 + a_2) + (-a_2 + a_3) + ..... + (-a_{n-1}+a_n) + (-a_n + a_{n+1}) = -a_0 + a_{n+1} = a_{n+1} - a_0$ are much more common. But no reason you can't have a "telescoping" serially cancelling product. This are called "telescoping" because... the fold up and collapse upon themselves like a telescope does. (It helps if you've ever seen a telescope of tubes within tubes so that it collapses into a tiny case.) – fleablood Feb 19 '21 at 06:24

1 Answers1

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This is an interesting observation. To see what's going on it helps to rewrite $(1 + 1/n)$ as $\frac{n+1}{n}$. Then for example if $x = 4$ we have $$ \Pi_{n=1}^4 (1 + \frac1n) = \frac21 \cdot \frac32 \cdot \frac 43 \cdot \frac54 = 5. $$ Everything cancels out except for the numerator in the final term in the product.

I'm not aware of any name for this formula.

littleO
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