pointwise convergent subsequences of $f_n(x)=(-x)^n, x\in [0,1]$

Question

I want to find its pointwise convergent sequence, if any.

$f_n$ itself converges pointwise everywhere, and there will be a pointwise convergent sequence of odd and even, so at least two.

Can you specify what you mean by "pointwise convergent sequence"? Sequence on $n$ for fixed $x$? Sequence on $x$ for fixed $n$? Sequences on $n,x$? — Jsevillamol, Dec 21 '16 at 09:11

score 6 · Answer 1 · answered Dec 21 '16 at 09:10

6

For $x\in[0,1)$, $f_n(x)=(-1)^nx^n\to0$ since $(-1)^n$ is bounded by $1$ and $x^n\to0$.

For $x=1$, $f_{2n}(x)=1$ and $f_{2n-1}(x)=-1$ so you do not have convergence at $x=1$.

answered Dec 21 '16 at 09:10

Landon Carter

ohh yes $f_n$ it self is not pointwise convergent everywhere as on $x=1$ – blabla Dec 21 '16 at 09:16

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