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I have this series:

$$\require{cancel} \sum_\limits{n=1}^\infty \ \dfrac{\sin(4n)}{4^n}$$

I tested out the Ratio Test for this one, and I don't know if the steps were right:

Ratio Test

$$\lim_\limits{n \to \infty} \vert \frac{\frac{\sin(4n+4)}{\cancel{4^n} \cdot \ 4}}{\frac{\sin(4n)}{\cancel{4^n}}} \vert$$ $$\lim_\limits{n \to \infty} \vert \frac{\sin(4n+4)}{4 \ \sin(4n)} \vert$$ The limit here oscillates between two values $0$, and $\infty$. I was a bit puzzled by it, and went to check the ratio test definition in my textbook. It says if $L = \infty$ then the series is divergent. I ended up going on Wolfram Alpha, and got this it converges.

It says it uses the comparison test, so I ended up doing the following:

Absolute Convergence Test

$$\sum_\limits{n=1}^\infty \frac{\vert \sin(4n) \vert}{4^n}$$

Comparison Test

I used the comparison test as directed, and did the following: $\frac{\vert \sin(4n) \vert}{4^n} \le \frac{1}{4^n} = (\frac{1}{4})^n$ By Geometric Series Test the other series converges, therefore my mystery series converges.

My Question:

Did the ratio test crash, or did I goof up in the using the ratio test?

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    $\begingroup$ The limit for the ratio test does not converge so the ratio test cannot be applied. One cannot say that the limit is $\infty$ as you seem to imply. $\endgroup$ Dec 31, 2019 at 18:35

1 Answer 1

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$(n,f(n))$ for $n=1,\dots,200$

The ratio test only works if the limit exists, however $f(n)=\frac{|\sin(4n+4)|}{4|\sin(4n)|}$ does not converge at infinity, therefore the test cannot be applied.

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