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I have the following series:

$\sum_n{a_n}$, where $a_1 = 1$, $a_{n+1} = a_n \,\frac{2 + \cos n}{\sqrt n}$. Does it converge or diverge?

Edit: formatted.

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    @DonAntonio You got it right! I did not know it was using LaTeX at all. May you help me to figure it out now?>2012-11-20

4 Answers 4

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Use the Ratio Test. You can read about it here.

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    Yep. If the limit of the ratio of a term to the previous term is less than 1, then the terms can be summed to a finite sum.2012-11-20
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Consider the sequence $a_{n+1}=a_n\frac{3}{\sqrt{n}}$. See whether it converges or not. Now see if this converges, why should the original sequence converge.

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    You have to see the limit for the ratio test. What is the ratio $|\frac{a_{n+1}}{a_n}|$ at the limit $n$ tends to infinity. Now see that each term of the original sequence is upper bounded by each term in this sequence.2012-11-20
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Continuing with Alex's answer: apply now Dirichlet's test*, with the monotone descending sequence $\,\displaystyle{\left\{\frac{1}{\sqrt n}\right\}}\,$ and the bounded sequence $\,\displaystyle{\left\{\sum_{k=1}^n\cos n\right\}}\,$

  • cazelais.disted.camosun.bc.ca/250/dirichlet.pdf
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Alternatively, use direct comparison with a geometric series, noting that all terms are positive. The first 15 terms sum to whatever they sum to. After that, with $n\ge16$, then $a_{n+1}=a_n\frac{2+\cos(n)}{\sqrt{n}}\le\frac{3}{4}a_n$. Inductively, $a_n\le\left(\frac34\right)^{n-16}a_{16}$ for $n\geq16$. So the series sums to at most $\sum_{n=1}^{15}a_n+a_{16}\sum_{n=16}^\infty\left(\frac34\right)^{n-16}$ or rather $\sum_{n=1}^{15}a_n+4a_{16}$