The convergence Criteria ratio

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The convergence criteria ratio indicates that if the limit of abs(an+1/an) or n√abs(an) is less than 1, the series is absolutely convergent. A limit equal to 1 suggests the series may be convergent but not necessarily absolutely convergent. The discussion emphasizes the importance of the limit value in determining the type of convergence. Clarification is provided that a limit less than 1 guarantees absolute convergence. Understanding these criteria is crucial for analyzing series convergence effectively.
Amaelle
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Homework Statement
is the serie convergent or absolutely convergent?
Relevant Equations
The convergence Criteria ratio
Greetings all

I have a question regarding the convergence criteria ratio, abs(an+1/an) or the n√abs(an) when the limit tend to a value less than 1 does it mean the serie is convergent or absolutely convergent?

Thank you!
 
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If the limit is less than 1, the series is absolutely convergent. It is only if the limit equals one that it might be convergent, but not absolutely convergent.
 
FactChecker said:
If the limit is less than 1, the series is absolutely convergent. It is only if the limit equals one that it might be convergent, but not absolutely convergent.
thank you!
 

Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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