Using comparison tests and limit comparison test

Thread starter Sunwoo Bae
Start date Nov 28, 2021
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Comparison Comparison test Convergence test Limit Series Test

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The series in question converges by the limit comparison test, as indicated by the answer sheet. It is also acceptable to use the comparison test for this specific problem. Both methods yield valid results for determining convergence. The discussion confirms that either approach can be applied effectively. Therefore, using the comparison test is indeed appropriate in this context.

Nov 28, 2021

Sunwoo Bae

Homework Statement: Determine whether the following series converge or diverge by using appropriate test. (Picture below)

Relevant Equations: None

The answer sheet states that the series converges by limit comparison test (the second way).
In the case of this particular problem, would it be also okay to use the comparison test, as shown above? (The first way)

Thank you!

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Nov 28, 2021

Office_Shredder

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Science Advisor

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Using the comparison test is fine here.

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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