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I have this problem here:

Given the series

[tex]\sum_{n=1} ^{\infty} \frac{x^{n+1}}{n(n+1)}[/tex]

show that this converges for every [tex]x \in \{ w \in \mathbb{C} \| \|w \| \leq 1 \}[/tex]

Solution:

Since

[tex] \sum_{n=1} ^{\infty} \left| \frac{x^{n+1}}{n(n+1)} \right| = \sum_{n=1} ^{\infty} \frac{1}{n(n+1)} \right| [/tex]

Since [tex]\sum_{n=1} ^{\infty} \frac{1}{n(n+1)} \right| [/tex] convergence then according the definition then

[tex]\sum_{n=1} ^{\infty} \frac{x^{n+1}}{n(n+1)}[/tex] is absolute convergent.

Thus

[tex]x \in \{ w \in \mathbb{C} \| \|w \| \leq 1 \}[/tex]. Right?

Best Regards

Fred

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# Absolute converge of a series

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