I need to find the sum of the series $\sum_{n=1}^{\infty} a_n$ where:
$$a_n=\Bigl(2\pi n+\frac{1}{n}\Bigr)\Bigl\lvert \sin\Bigl(\frac{1}{n}\Bigr)\Bigr\rvert$$
as a part of a disproof. I think the limit is $2\pi$ but I'm stuck. Any help?
Finding a limit for a disproof
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$\begingroup$
sequences-and-series
limits
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0The limit of $a_n$ is simple to find and you got it. Waht about $\sum_{n=1}^{\infty} a_n$ – 2017-01-14
2 Answers
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write you term in the form $$\left(2\pi+\frac{1}{n^2}\right)\left|\frac{\sin(\frac{1}{n})}{\frac{1}{n}}\right|$$ and we get the limit as $$2\pi$$
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Hint : Use $$\lim_{n\rightarrow\infty} \frac{|\sin(\frac{1}{n})|}{\frac{1}{n}}=1$$