My teacher said $1\over n$ is not element of $\ell^1$ .I know if $\sum_{i=1}^\infty|{1\over n}| < \infty$ , it is element of $\ell^1$. Let's consider the series $(x_i)=(1,{1\over 2}, {1\over 3} , . . .)$ .It seems to me if i sum all the component of the series up , it will be smaller than $2$ so it is finite. Is it right ? Thank you for your help.
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5$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{25}{12} > 2$. – Unit Mar 27 '15 at 19:17
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2Look up "harmonic series". – Robert Israel Mar 27 '15 at 19:19
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2I got it . Thank you. – izaag Mar 27 '15 at 19:22