Hi(adsbygoogle = window.adsbygoogle || []).push({});

Looking at the series

[tex]\sum \limit_{n=1} ^{\infty} \frac{z^{n+1}}{n(n+1)}[/tex]

This series has the radius of Convergence R = 1.

Show that the series

converge for every [tex]z \in \{w \in \mathbb{C} | |w| \leq 1 \}[/tex]

And Secondly I need to show that

[tex]g(z) = \sum \limit_{n=1} ^{\infty} \frac{z^{n+1}}{n(n+1)}[/tex]

Is continius in [tex]z \in \{w \in \mathbb{C} | |w| \leq 1 \}[/tex]

Solution:

(1)

Since R = 1, then

[tex]\displaystyle \lim_{n \rightarrow \infty} b_n = \displaystyle \lim_{n \rightarrow \infty} \frac{1}{n(n+1)} = 0[/tex]

[tex]b_n = \displaystyle \lim_{n \rightarrow \infty} b_n = \displaystyle \lim_{n \rightarrow \infty} \frac{1}{(n+1)(n+1)+1} = b_{n +1} [/tex]

Therefore converge the [tex]z \in \{w \in \mathbb{C} | |w| \leq 1 \}[/tex]

(2) Doesn't that follow from (1) ?

Sincerely Yours

Hummingbird25

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: QUESTION: Series

**Physics Forums | Science Articles, Homework Help, Discussion**