MHB Summing a Series of Cubic Roots

anemone
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Sum the series below:

$$\sum_{n=1}^{999}\dfrac{1}{a_n}$$ where $a_n=\sqrt[3]{n^2-2n+1}+\sqrt[3]{n^2+2n+1}+\sqrt[3]{n^2-1}$.
 
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Here is my solution.

Note

$$a_n = (n-1)^{2/3} + (n + 1)^{2/3} + (n-1)^{1/3}(n+1)^{1/3} = \frac{(n+1) - (n-1)}{\sqrt[3]{n+1} - \sqrt[3]{n-1}} = \frac{2}{\sqrt[3]{n+1} - \sqrt[3]{n-1}}.$$

Therefore

$$\sum_{n = 1}^{999} \frac{1}{a_n} = \frac{1}{2}\sum_{n = 1}^{999} (\sqrt[3]{n+1} - \sqrt[3]{n-1}) = \frac{1}{2}\sum_{n = 1}^{999} [(\sqrt[3]{n+1} - \sqrt[3]{n}) + (\sqrt[3]{n} - \sqrt[3]{n-1})]$$
$$ = \frac{1}{2}[(\sqrt[3]{1000} - \sqrt[3]{1}) + (\sqrt[3]{999} - \sqrt[3]{0})] = \frac{9 + 3\sqrt[3]{37}}{2}.$$
 
Euge said:
Here is my solution.

Note

$$a_n = (n-1)^{2/3} + (n + 1)^{2/3} + (n-1)^{1/3}(n+1)^{1/3} = \frac{(n+1) - (n-1)}{\sqrt[3]{n+1} - \sqrt[3]{n-1}} = \frac{2}{\sqrt[3]{n+1} - \sqrt[3]{n-1}}.$$

Therefore

$$\sum_{n = 1}^{999} \frac{1}{a_n} = \frac{1}{2}\sum_{n = 1}^{999} (\sqrt[3]{n+1} - \sqrt[3]{n-1}) = \frac{1}{2}\sum_{n = 1}^{999} [(\sqrt[3]{n+1} - \sqrt[3]{n}) + (\sqrt[3]{n} - \sqrt[3]{n-1})]$$
$$ = \frac{1}{2}[(\sqrt[3]{1000} - \sqrt[3]{1}) + (\sqrt[3]{999} - \sqrt[3]{0})] = \frac{9 + 3\sqrt[3]{37}}{2}.$$

Thanks Euge for participating in this IMO Problem from China!

For me, it took me a fraction of time to realize $a_n$ is a geometric series that consists of the first three terms. :)
 
Hi anemone,

What do you mean when you say that $a_n$ is a geometric series? There isn't a common ratio between consecutive terms.
 
Euge said:
Hi anemone,

What do you mean when you say that $a_n$ is a geometric series? There isn't a common ratio between consecutive terms.

Hi Euge,
If I rewrite $a_n$ in such a way that it now becomes $a_n=\sqrt[3]{n^2+2n+1}+\sqrt[3]{n^2-1}+\sqrt[3]{n^2-2n+1}$, then I noticed that

$\dfrac{\sqrt[3]{n^2-1}}{\sqrt[3]{n^2+2n+1}}=\dfrac{\sqrt[3]{n^2-2n+1}}{\sqrt[3]{n^2-1}}$

Therefore, $\sqrt[3]{n^2+2n+1}+\sqrt[3]{n^2-1}+\sqrt[3]{n^2-2n+1}$ is a geometric series with:
  • the first term $\sqrt[3]{n^2+2n+1}=(n+1)^{\frac{2}{3}}$ and
  • the common ratio of $\left(\dfrac{n-1}{n+1}\right)^{\frac{1}{3}}$.

Therefore, $a_n=\dfrac{((n+1)^{\frac{2}{3}})\left(1-\left(\left(\dfrac{n-1}{n+1}\right)^{\frac{1}{3}}\right)^3\right)}{1-\left(\dfrac{n-1}{n+1}\right)^{\frac{1}{3}}}=\dfrac{2}{(n+1)^{\frac{1}{3}}-(n-1)^{\frac{1}{3}}}$
 
Hi anemone,

Thanks for clarifying. Now I understand what you mean. (Nod)
 
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