What's the pointwise convergence of $\{nx^n(1-x)\}$?

Question

Is $\{nx^n(1-x)\}_{n=1}^{\infty}$ pointwise convergent if $x \in \mathbb{R}$? If it is, what's the pointwise limit?

See this also: http://math.stackexchange.com/questions/55468/how-to-prove-that-exponential-grows-faster-than-polynomial — Seven, Dec 12 '15 at 06:07

score 1 · Accepted Answer · answered Dec 12 '15 at 06:26

$f_n(x)=nx^n(1-x)$ ;

Note that $\lim_{n\to \infty} x^n=0\iff |x|<1\implies \lim_{n\to \infty}f_n(x)=0$.

Also if $x=1\implies f_n(x)=0\forall n\implies \lim_{n\to \infty} f_n(x)=0.$

However if $x=-1$ or $|x|>1\implies \lim_{n\to \infty} f_n(x)=\infty$

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