Check for the convergence or divergence of the following series

  • #1
freshman2013
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Homework Statement


Here are some series I'm completely stuck on.
1.sqrt(n)*(1-cos(1/n))

2. a series in which if n is odd, then an is 1/(n+[itex]\sqrt[]{n}[/itex]) while if n is even, then an is -1/n


Homework Equations





The Attempt at a Solution


For 1., I tried integral test which seemed impossible to integrate, and then I tried comparison test but i can't find anything to compare to

For 2. I thought that as n approaches infinity for 1/(n+[itex]\sqrt[]{n}[/itex]), which equals (1/[itex]\sqrt{n}[/itex])/([itex]\sqrt{n}[/itex]+1), then that expression pretty much looks like 1/n and thus the series is approx (-1)^n/n and is thus conditionally convergent like (-1)^n/n. But I then realized perhaps you can't apply the alternating series test since in the original series, the abs value of it is not decreasing for every term.E.g.1/(100+root(100))=1/110 < 1/101 Anyone have any idea on how to tackle this, or am I right from my original attempt?
 
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Answers and Replies

  • #2
STEMucator
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Homework Statement


Here are some series I'm completely stuck on.
1.sqrt(n)*(1-cos(1/n))

2. a series in which if n is odd, then an is 1/(n+[itex]\sqrt[]{n}[/itex]) while if n is even, then an is -1/n


Homework Equations





The Attempt at a Solution


For 1., I tried integral test which seemed impossible to integrate, and then I tried comparison test but i can't find anything to compare to

For 2. I thought that as n approaches infinity for 1/(n+[itex]\sqrt[]{n}[/itex]), which equals (1/[itex]\sqrt{n}[/itex])/([itex]\sqrt{n}[/itex]+1), then that expression pretty much looks like 1/n and thus the series is approx (-1)^n/n and is thus conditionally convergent like (-1)^n/n. But I then realized perhaps you can't apply the alternating series test since in the original series, the abs value of it is not decreasing for every term.E.g.1/(100+root(100))=1/110 < 1/101 Anyone have any idea on how to tackle this, or am I right from my original attempt?

Let's start with #1.

Your series is: ##\sum_{n=1}^{∞} \sqrt{n} - \sqrt{n}cos(\frac{1}{n})##

I know this may seem misleading, but this series appears to converge.

Hint: ##|cos(\frac{1}{n})| ≤ 1##
 
  • #3
haruspex
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Let's start with #1.
I know this may seem misleading, but this series appears to converge.

Hint: ##|cos(\frac{1}{n})| ≤ 1##
Yes, it converges, but you need to use a stronger fact about cos of small angles. Freshman2013, do you know the Taylor expansion of cos?
 
  • #4
jbunniii
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For the first problem, this trig identity may be useful: ##2\sin^2(x) = 1 - \cos(2x)##.
 
  • #5
jbunniii
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For problem 2, try adding ##a_n## and ##a_{n+1}##, where ##n## is odd.
 
  • #6
freshman2013
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so for the second one, could I do this:
the first term 1/(n+[itex]\sqrt{n}[/itex]) for all odd n, I rewrite as a separate series 1/(2n-1+[itex]\sqrt{2n-1}[/itex]) and the 2nd term -1/n for all even n I rewrite as -1/(2n)

I combine those two series and I get (1-[itex]\sqrt{2n-1}[/itex])/(4n^2-2n+2n[itex]\sqrt{2n-1}[/itex]) and test for that(which is convergent). And I do the same except with 1/2n instead -1/2n to check for absolute convergence.
 
  • #7
jbunniii
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And I do the same except with 1/2n instead -1/2n to check for absolute convergence.
For absolute convergence, you can do something simpler. I claim that if there is some subsequence ##(n_k)_{k=1}^{\infty}## such that ##\sum_{k=1}^{\infty} |a_{n_k}|## diverges, then ##\sum_{n=1}^{\infty} |a_n|## must also diverge. If you can prove this, then you can apply it immediately to your series.
 

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