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I want to ask for two examples in the following cases:

1) Given a bounded sequence $\{a_n\}$, $\lim_{n\to \infty}{(a_{n+1}-a_n)}=0$ but $\{a_n\}$ diverges.

2) A function defined on real-line $f(x)$'s Taylor series converges at a point $x_0$ but does not equal to $f(x_0)$.

Thanks for your help.

Edit

in 2), I was thinking of the Taylor series of the function $f$ at the point $x_0$.

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    @Arturo: As I said, it's a linguistic issue, not a mathematical one. I do understand your point of view, and I will conform to your "rules" as long as I will post here. About the comments, never mind (the one I didn't like has been deleted); I'm sure the guys will understand and will try to be less sarcastic next times.2011-03-28

5 Answers 5

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Eric gave an example of 2). As regards 1), let $a_n=\cos(2\pi k/2^i)$ if $n$ is between $2^i$ and $2^{i+1}$ and $n=2^i+k$ with $0\le k\le 2^{i+1}-1$. Then $|a_{n+1}-a_n|\le2\pi/2^i\le4\pi/n$ hence $a_{n+1}-a_n\to0$ but the limit set of the sequence $(a_n)$ is $[-1,1]$.

Another example is $a_n=\cos(\log n)$. Then $|a_{n+1}-a_n|\le1/n$ and the limit set is $[-1,1]$.

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    @Eric Yup. @Alon had the same idea.2011-03-27
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For 1): $ 1,\frac12,0,\frac13,\frac23,1,\frac34,\frac24,\frac14,0,\frac15,\frac25,\frac35,\frac45,1,\frac56,\frac46,\frac36,\frac26,\frac16,0,\frac17,\dots $ I leave it to you to find an explicit formula for $a_n$.

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    Very nice indeed!2011-03-27
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2) Define the function $f$ as follows: $f(x)=e^{-1/x^2}\ \text{if } x>0$ and $f(x)=0\ \text{if } x\leq 0.$ Now consider the Taylor series centered at zero. This provides an example of a non-analytic $C^\infty$ function. This taylor series converges everywhere, but is identically zero, and $f(x)$ is not identically zero.

Hope that helps,

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    @Qiang: You can't of meant this? The Taylor series centered at $x_0$ always agrees with the function at the point $x_0$, by _definition_ of Taylor series. Didier hints at this in his comment.2011-03-27
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2) It is classical:

$f(x)=\begin{cases} \exp (-\tfrac{1}{x}) &\text{, if } x>0 \\ 0 &\text{, if } x\leq 0 \end{cases}$

and $x_0=0$.

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    @Qiang, you had the same comment, so I deleted mine.2011-03-27
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Hint for 1): Find a sequence $t_n$ which tends to infinity with $t_{n+1}-t_n \to 0$ as $n \to \infty$, and sample a periodic function such as $\sin(x)$ at the points $t_n$.