Radius of convergence

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rettore84

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Hello to all,

I'm trying to solve the question attached. The answer given is "e" however I'm getting 1. Could someone please help me? Thanks in advance.
 

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rettore84 said:
Hello to all,

I'm trying to solve the question attached. The answer given is "e" however I'm getting 1. Could someone please help me? Thanks in advance.
Click to expand...

attachment.php


I cannot read your problem statement!
 
I'm sorry for the low quality image. The series is (n!/n^n)*x^n. Attached is a new picture, please see if this one is better.
 

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rettore84 said:
Hello to all,

I'm trying to solve the question attached. The answer given is "e" however I'm getting 1. Could someone please help me? Thanks in advance.
Click to expand...

You are all right down to
\(\displaystyle \underset{n \to \infty}{lim}\, \frac{n^n\, x}{(n+1)^n}\)
However, you can not just cross out the 1 on the n+1. Depending on just what you have studied, you should be able to get to
\(\displaystyle \underset{n \to \infty}{lim}\, (\frac{n+1}{n})^n\, =\,\underset{n \to \infty}{lim}\, (1\, +\, \frac{1}{n})^n\, =\, e\)
 
You are all right down to
[FONT=MathJax_Math]l[/FONT][FONT=MathJax_Math]i[/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math]n[/FONT]
However, you can not just cross out the 1 on the n+1. Depending on just what you have studied, you should be able to get to
[FONT=MathJax_Math]l[/FONT][FONT=MathJax_Math]i[/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]l[/FONT][FONT=MathJax_Math]i[/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]e[/FONT]
Click to expand...

ok I got it that this [FONT=MathJax_Math]l[/FONT][FONT=MathJax_Math]i[/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math]n[/FONT] can be --> lim (n/n+1)^n and this converges to "1/e", because this is the reciprocal of this lim (n+1/n)^n which is "e".

So the answer should be 1/e?
[FONT=MathJax_Math]l[/FONT][FONT=MathJax_Math]i[/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main][/FONT][FONT=MathJax_Math]l[/FONT][FONT=MathJax_Math]i[/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Math]l[/FONT][FONT=MathJax_Math]i[/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math]n[/FONT]
 
You're on the right track. The limit does converge to 1/e, but I don't think that's the answer your book/instructor is looking for. You didn't post the instructions, so I'm just guessing here, but based on the title of your post, the given task was to find the radius of convergence. So, find the values of x where the expression is less than one:

\(\displaystyle \displaystyle \lim _{n\to \infty }\left(\left|\frac{x\cdot n^n}{\left(n+1\right)^n}\right|\right)= \left|x\right|\cdot \lim _{n\to \infty }\left(\left(\frac{n}{n+1}\right)^n\right)=\left|x\right|\cdot \frac{1}{e}\)

\(\displaystyle \left|x\right|\cdot \frac{1}{e}<1\)

You take it from here...
 
Thank you ksdhart, you're right I forgot that I was looking for the radius of convergence and should make the limit < 1. Thanks!:D
 
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