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I want to expand and test this $\{(n^3+1)^{1/3} - n\}$ for convergence/divergence.

The edited version is: Test for convergence $\sum_{n=1}^{\infty}\{(n^3+1)^{1/3} - n\}$

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    Well! I am editing the question. I think I got the answer from avatar BUT I still have a little bit of doubt about his answer. I am writing the doubt as a comment to his answer2012-06-29

3 Answers 3

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The first thing we note is that, as $n\to \infty$ the term $(n^3+1)^{1/3}$ is of order $n$, which is cancelled by the other term, hence we must ask about the next order approximation.

Now, $ (n^3+1)^{1/3} = n \left(1 + \frac{1}{n^3}\right)^{1/3}$ and $(1+x)^{1/3} = 1 + \frac{1}{3}x + O(x^2)$, hence

$ (n^3+1)^{1/3} - n = \frac{1}{3}\frac{1}{n^2}+O(n^{-5})$

Hence, the series converge.

(If you want an explicit bound, you can just use $ (1+x)^{1/3} < 1+x$ for $x>0$, so you can compare to the series $\sum 1/n^2$ which you're supposed to know is convergent)

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Let $y=1/n$, then $\lim_{n\to \infty}{(n^3+1)}^{1/3}-n=\lim_{y\to0^+}\frac{(1+y^3)^{1/3}-1}{y}$. Using L'Hopital's Rule, this limit evaluates to $0$.Hence, the expression converges. You can also see that as $n$ increases, the importance of $1$ in the expression ${(n^3+1)}^{1/3}-n$ decreases and $(n^3+1)^{1/3}$ approaches $n$.Hence, the expression converges to $0$(as verified by limit). For the series part as @siminore pointed out,this difference ${(n^3+1)}^{1/3}-n$ is of the order of $1/n^2$,therefore, the sum of these is of the order of this sum :$\sum_{1}^{\infty}1/n^2$ which is $= {\pi}^2/6$. Thus the series is bounded and hence converges .

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    For summation part visit http://math$2$.org/math/expansion/power2.htm2012-06-29
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By direct inspection, for every pair of real numbers $A$ and $B$, $ A^3 - B^3 = (A-B) (A^2+AB+B^2). $ Choose now $A=\sqrt[3]{n^3+1}$ and $B=n$. Then $ (n^3+1)^{1/3} - n = \frac{n^3+1-n^3}{(n^3+1)^{2/3} + n (n^3+1)^{1/3}+n^2} \sim \frac{1}{n^2} $ as $n \to +\infty$. The limit is therefore zero.

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    We will never agree on the use of symbols :-) I learned that $a_n \approx b_n$ means that $a_n / b_n \to 1$, and that $a_n \sim b_n$ that $a_n / b_n$ tends to a limit different than zero (and finite). It seems hard to find a set of symbols that everybody can understand.2012-06-29