anonymous
  • anonymous
what is limit of sequence...?
Mathematics
katieb
  • katieb
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anonymous
  • anonymous
Let \[\varepsilon > 0\] be arbitrary. Then \[\exists N \in \mathbb{N} : |a_n - l| < \varepsilon \hspace{0.3cm} \forall n > N\] There is a dependence of \[\varepsilon\] on \[N\] and it's written as \[\varepsilon = \varepsilon (N).\] In the above, \[a_n\] are just terms of your sequence for \[n>N.\]
anonymous
  • anonymous
\[l\] being the limit of your sequence. If you are given that \[l\] exists for your sequence, then the above is true by definition.
anonymous
  • anonymous
actually more properly, \[N=N(\varepsilon )\] is the dependence as epsilon is arbitrary.

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