It's a product? Not a sum?

Calculus Level 4

$\large \displaystyle \lim_{n \to \infty} \prod_{r=3}^{n} \dfrac{r^3 - 2^3}{r^3 + 2^3}$

If the above limit can be expressed in the form $\dfrac{a}{b}$ where $a$ and $b$ are coprime positive integers, find the value of $a+b$ .

The answer is 9.

1 solution

First Last
Apr 18, 2017

$\displaystyle\prod_{r=3}^n\frac{r^3-8}{r^3+8} = \prod_{r=3}^n\frac{(r-2)(r^2+2r+4)}{(r+2)(r^2-2r+4)}$ As n approaches infinity, the terms in each product cancel out leaving:

$\displaystyle\frac{1}{5}\times\frac{2}{6}\times\frac{3}{7}\times\frac{4}{8}\times\frac{5}{9}\times\frac{6}{10}\times...\quad\times\frac{19}{7}\times\frac{28}{12}\times\frac{39}{19}\times\frac{52}{28}\times... = \boxed{\frac{2}{7}}$

It's a product? Not a sum?

The answer is 9.

1 solution

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