Let $\{v_n\}_{n \in \Bbb N} \in \ell^2$ be a sequence of $\ell^2$ over $\Bbb C$ such that $\lim_{n \to \infty} v_n = 0 $
I would like to know if is true the following relation:
$\sum_{n=1}^\infty v_n = 0 $
Thanks
Sequence in $\ell^2$ converget to $0$ if it is summable
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sequences-and-series
complex-analysis
functional-analysis
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0What if $v_1 = 1$ and all the other $v_i = 0$? – 2017-02-01
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0@EthanBolker let $v=(1,0,...)$ do you mean this? – 2017-02-01
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0Or any sequence with only a finite number of nonzero terms. If you're more ambitious, take the terms of a convergent geometric series with positive ratio $r <1.$ – 2017-02-01
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0No thoughts or work on this problem? – 2017-02-01
1 Answers
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No, this is not true. Take the series $\nu_n=\frac{1}{n}$.
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0my $v_n$ isn't a sequence of complex number but a sequence of $\ell^2$ or i don't understand you? – 2017-02-01
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1Every real number is also complex, so my $\nu_n$ is an element of $\ell ^2$: the series $\sum_{n=1}^\infty \frac{1}{n^2}$ converges. – 2017-02-01
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0ok thansk.. i have wiswrotten – 2017-02-01