I need to proof convergence, and find the radius of converge

asi123
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Homework Statement



Hey guys, can you help me with this on please?
First one, I need to proof convergence, and the second one is to find the radius of converge.

Homework Equations





The Attempt at a Solution

 

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For the first one, you are given the first series converges. That means a_n->infinity. Think comparison test. 1/(a_n-x)<1/(a_n-y) if x<y<a_n. How about choosing y=a_n/2?? Can you justify that? For the second one, write it as [(n+1)^n/n^n]*[1/n^z]. The first factor has a limit. What is it?
 
Dick said:
For the first one, you are given the first series converges. That means a_n->infinity. Think comparison test. 1/(a_n-x)<1/(a_n-y) if x<y<a_n. How about choosing y=a_n/2?? Can you justify that? For the second one, write it as [(n+1)^n/n^n]*[1/n^z]. The first factor has a limit. What is it?

Right, it's e. so that's mean that in order for the series to converge, x need to bigger then 1?
 
Looks to me like aside from the e, it's a power series.
 
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Dick said:
Looks to me like aside from the e, it's a power series.

This are my thoughts, is this right?
 

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asi123 said:
This are my thoughts, is this right?

Why do you think 1/(a_n-2) converges? Shouldn't you state a reason?
 
Dick said:
Why do you think 1/(a_n-2) converges? Shouldn't you state a reason?

If 1/(a_n-2) converges, than why shouldn't 1/(a_n-2) converge? I mean if a_n -> infinity than I don't think that 2 will bother him, no?
 
asi123 said:
If 1/(a_n-2) converges, than why shouldn't 1/(a_n-2) converge? I mean if a_n -> infinity than I don't think that 2 will bother him, no?

No, the 2 won't bother him. But you still have to show that. Set up a comparison test with something you know converges. Review my hint about this one.
 
Dick said:
No, the 2 won't bother him. But you still have to show that. Set up a comparison test with something you know converges. Review my hint about this one.

Yeah, I got you, thanks.
 
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