An algebra problem by kellen atkins

Algebra Level 3

What is

1 + 1 1 ( 1 + 1 2 ( 1 + 1 3 ( 1 + 1 4 ( 1 + 1 5 ( ⋯ ) ) ) ) ) ? 1+\frac{1}{1}\left(1+\frac{1}{2}\left(1+{\small\frac{1}{3}\Big(1+\tfrac{1}{4}\big(1+\tfrac{1}{5}(\cdots )\big)\Big)}\right)\right)?


The answer is 2.71828182846.

Kellen Atkins by kellen atkins , 19, USA

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1 solution

Chew-Seong Cheong
Sep 16, 2018

Let the number be x x . Then we have:

x = 1 + 1 1 ( 1 + 1 2 ( 1 + 1 3 ( 1 + 1 4 ( 1 + 1 5 ( ⋯ ) ) ) ) ) = 1 + 1 1 ! + 1 2 ! + 1 3 ! + 1 4 ! + 1 5 ! + ⋯ = e ≈ 2.718 \begin{aligned} x & = 1 + \frac 11\left(1+\frac 12\left(1+\frac 13\left(1+\frac 14\left(1+\frac 15 \bigg( \cdots \bigg) \right) \right) \right) \right) \\ & = 1 + \frac 1{1!} + \frac 1{2!} + \frac 1{3!} + \frac 1{4!} + \frac 1{5!} + \cdots \\ & = e \approx \boxed{2.718} \end{aligned}

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