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I have problems with two exercises. I know the answer of those two but both of them have a step which I don't understand.

1) I have to prove that $\prod_{k=2}^n \frac{k^3-1}{k^3+1}$ is convergent.

My first steps were:

$\prod_{k=2}^n \frac{k^3-1}{k^3+1}= \prod_{k=2}^n \frac{k-1}{k+1}\cdot \prod_{k=2}^n \frac{k^2+k+1}{k^2-k+1}=\frac{2(n-1)!}{(n+1)!}\prod_{k=2}^n \frac{k^2+k+1}{k^2-k+1}$

And then I got stuck. The author of the exercise does that:

$\frac{2(n-1)!}{(n+1)!}\cdot\prod_{k=2}^n \frac{k^2+k+1}{k^2-k+1} = \frac{2}{n(n+1)}\cdot\prod_{k=2}^n((k+1)^2-(k+1)+1)\cdot\prod_{k=2}^n\frac{1}{k^2-k+1}=$ $\frac{2}{n(n+1)}\cdot\prod_{k=3}^{n+1}(k^2-k+1)\cdot\prod_{k=2}^n\frac{1}{k^2-k+1}=\frac{2}{n(n+1)}\cdot\frac{(n+1)^2-(n+1)+1}{3}=\frac{2}{3}\cdot\frac{n^2+n+1}{n^2+n}\rightarrow\frac{2}{3}$

I understand what he does, but I don't know how he knew it. How can I use it for other exercises, so does somebody know how to remember this trick and for which kind of exercises this works?

2) I have to prove that $a_n:=\sqrt{n^2+n}-n$ is convergent and that a constant $A$ exist with $|a_n-a|<\frac{A}{n}.$

I proved that $a_n$ is convergent to $\frac{1}{2}$. For finding the constant $A$ the author says

It's easy to see, that: $|a_n -\frac{1}{2}|=|\frac{n}{\sqrt{n^2+n}+n}-\frac{1}{2}|<\frac{1}{8n}$, therefore $A:=\frac{1}{8n}$.

Unfortunately I don't see it. Can somebody please give a hint how he gets $\frac{1}{8}$?

thanks

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*Therefore $A=\frac18$* and not $A=\frac1{8n}$. –  Did Sep 25 '11 at 10:54
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2 Answers

1) Factoring further $$ \begin{align} k^3-1&=(k-1)(k^2+k+1)=(k-1)(k+\alpha)(k+\bar{\alpha})\\ k^3+1&=(k+1)(k^2-k+1)=(k+1)(k-\alpha)(k-\bar{\alpha}) \end{align} $$ where $\alpha+\bar{\alpha}=1$. It is easy to see that $\prod\frac{k-1}{k+1}$ is telescoping. However, when we write $\prod\frac{(k+\alpha)(k+\bar{\alpha})}{(k-\alpha)(k-\bar{\alpha})}=\prod\frac{(k+1-\bar{\alpha})(k+1-\alpha)}{(k-\alpha)(k-\bar{\alpha})}$, it becomes evident that this telescopes, too.

2) $$ \begin{align} \left|\frac{n}{\sqrt{n^2+n}+n}-\frac{1}{2}\right| &=\frac{1}{2}\left|\frac{n-\sqrt{n^2+n}}{n+\sqrt{n^2+n}}\right|\\ &=\frac{1}{2}\frac{n}{(n+\sqrt{n^2+n})^2}\\ &<\frac{1}{2}\frac{n}{4n^2}\\ &=\frac{1}{8n} \end{align} $$

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1) The author uses a trick to compute the exact value of the infinite product but one does not need it to show the convergence only.

Namely, use that $\frac{k^3-1}{k^3+1}=1-\frac2{k^3+1}$, that $\frac2{k^3+1}<1$ for every $k\ge2$ and that the series $\sum\limits_{k}\frac2{k^3+1}$ converges absolutely.

2) Use the trick of the conjugate quantity, that is, $\sqrt{x}-\sqrt{y}=\frac{x-y}{\sqrt{x}+\sqrt{y}}$ for positive $x$ and $y$.

Thus, $a_n=\frac{n}{\sqrt{n^2+n}+n}$ (choose $x=n^2+n$ and $y=n^2$) hence $1-2a_n=\frac{\sqrt{n^2+n}-n}{\sqrt{n^2+n}+n}$, which is also $1-2a_n=\frac{n}{(\sqrt{n^2+n}+n)^2}$ (same $x$ and $y$). Since $\sqrt{n^2+n}+n\ge2n$, this shows that $0\le 1-2a_n\le\frac1{4n}$, hence $2a_n\to1$ and, as the author writes, $\frac12-\frac1{8n}\le a_n\le\frac12$.

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