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I know that the sequence $$ f_n(x) = x^n $$ Converges pointwise to $$ f(x) = \begin{cases} 0 & 0\leq x < 1 \\ 1 & x=1\end{cases} $$ in $[0,1]$.

My question is - is it right to say that the sequence converges to the line $x = 1$ for $x\in(1,\infty)$?

Edit: Confused with sequence not series

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  • $\begingroup$ Yes I mixed another exercise I had, sorry. $\endgroup$ – JonTrav Jun 13 '15 at 7:11
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If $x>1$, the sequence $x^n$ diverges.

The line $x=1$ is always going to be distance 1 away from the point $(2,2^n)$. So there's no uniformity.

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  • $\begingroup$ I know that, but when graphing $x^n$ for $x>1$ as $n$ gets larger the graph gets more and more perpendicular to the $x$-axis. I know that it does not converge to any function, I asked about a curve $\endgroup$ – JonTrav Jun 13 '15 at 7:13
  • $\begingroup$ These kinda of convergence behaviors are usually measured with the hausdorf metric. I don't think it will converge in the hausdorf metric though. $\endgroup$ – Zach Stone Jun 13 '15 at 7:14

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