# Absolute convergence with Alternating series test

I was studying and stumbled across a question that I got right but on shaky basis.

The question was :

Let $$a_n = (-1)^nsin(\frac{1}{n^2})$$ with various answers possible. With the alternating series test, it was easy to say that $$\sum a_n$$ converge but there was two possible answers, one that it was only convergent and another that it was absolutely convergent.

I kind of guess it's absolutely convergent since the ratio test would be "infinitely slightly" below one but I'm not sure about how to do it mathematically or wether I got it right purely by luck...

• Hint: $|\sin x|<|x|$.
– user1010241
Jan 10, 2022 at 11:16
• oh well for some reason i didn't thought of it thanks Jan 10, 2022 at 11:59

It is absolutely convergent as well.
Consider this,
$$|a_n| = |sin(\frac{1}{n^2})| \leq \frac{1}{n^2}$$
I use the property: $$sin(\theta) \leq \theta$$
We know that $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ converges.
So, from comparison test, given series will also converge absolutely.

Hint:

Doubling the number of terms each time,

$$1+\frac14+\frac19+\frac1{16}+\frac1{25}+\frac1{36}+\frac1{49}+\cdots\frac1{n^2}+\cdots\\< 1+\frac14+\frac14+\frac1{16}+\frac1{16}+\frac1{16}+\frac1{16}+\cdots\underbrace{\frac1{2^{2k}}+\frac1{2^{2k}}+\cdots+\frac1{2^{2k}}}_{2^k}+\cdots\\<1+\frac24+\frac4{16}+\cdots\frac1{2^k}+\cdots<2.$$

Note that the same argument works for convergence of $$\dfrac1{n^{1+\epsilon}}$$ with $$\epsilon>0$$. And with a small modification, divergence of $$\dfrac1{n}$$.

• I don't really understand how I could use that, because yeah I can get a bound for $\frac{1}{n^2}$ but it will not tell me that $sin(\frac{1}{n^2})$ converge since sin(2) is just a constant. Jan 10, 2022 at 12:56
• @user15757055: this is a bound for the absolute series !
– user1010241
Jan 10, 2022 at 14:13
• Oh yeah sorry I don't know why it didn't clicked but it's definitely a proof that the series is absolutely convergent when you have a bound for it ahah sorry again and thanks for your help Jan 10, 2022 at 14:44