- #1

Euklidian-Space

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## Homework Statement

show ##\{nx^{n}(1-x)\}## converges uni on [0,1]

## Homework Equations

## The Attempt at a Solution

Note, that ##\sum nx^n(1-x)## converges uni on [0,1] by Abels' Theorem. Therefore the series follows the cauchy criterion,

##\forall \epsilon > 0##, ##\exists N > 0 ## such that if n,n > N then

$$\left|(m+1)x^{m+1}(1 - x) + ... + nx^{n}(1- x)\right| < \epsilon\,\,\, \forall x \in [0,1]$$,

Now since ##nx^n(1 - x) \geq 0## for all n we can say

$$\left|nx^{n}(1 - x) - mx^{m}(1 - x)\right| < \epsilon\,\,\, \forall x \in [0,1].$$

Which implies that ##\{nx^n(1 - x)\}## is uniformly cauchy ##\forall x \in [0,1]## and therefore uniformly convergent for ##x \in [0,1]##

Dont know if this is right