Check for the convergence or divergence of the following series

[freshman2013]

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 a_n is 1/(n+\sqrt[]{n}) while if n is even, then a_n 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+\sqrt[]{n}), which equals (1/\sqrt{n})/(\sqrt{n}+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?

 

[STEMucator]

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 a_n is 1/(n+\sqrt[]{n}) while if n is even, then a_n 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+\sqrt[]{n}), which equals (1/\sqrt{n})/(\sqrt{n}+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: $|cos(\frac{1}{n})| ≤ 1$

 

[haruspex]

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?

 

[jbunniii]

For the first problem, this trig identity may be useful: $2\sin^2(x) = 1 - \cos(2x)$.

 

[jbunniii]

For problem 2, try adding $a_n$ and $a_{n+1}$, where $n$ is odd.

 

[freshman2013]

so for the second one, could I do this:
the first term 1/(n+\sqrt{n}) for all odd n, I rewrite as a separate series 1/(2n-1+\sqrt{2n-1}) and the 2nd term -1/n for all even n I rewrite as -1/(2n)

I combine those two series and I get (1-\sqrt{2n-1})/(4n^2-2n+2n\sqrt{2n-1}) and test for that(which is convergent). And I do the same except with 1/2n instead -1/2n to check for absolute convergence.

 

[jbunniii]

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.