Alternating Series Doubts

Question

I've some doubts on Alternating series and Alternating test series. I was trying to clear my mind by practicing with some of my book exercises and found troubles with this one:

$$\sum_{n=0}^\infty \frac{(-1)^{n+1}}{\sqrt[n+1] {10}}$$

My logic is as follows:

The series doesn't converge since my $a_n = - \frac{1}{\sqrt[n+1]{10}}$ approaches $1$ as $n$ goes to $\infty$ and is $decreasing$ because of $\sqrt[n+1] {10} > \sqrt[n+2] {10}$

I don't want to stop here so my question: is there a way to know if this series diverges to $+\infty$ or $-\infty$?

Answer 1

First, it is very important in mathematics to carefully read and understand the definitions that are in play. In this case, the following are probably relevant:

Definition: Let $\{a_n\}$ be a sequence of real numbers. We say that the series $$ \sum_{n=1}^{\infty} a_n $$ converges if the sequence of partial sums $$ S_N := \sum_{n=1}^{N} a_n $$ converges. Oherwise, we say that the series diverges.

That is, understanding the behaviour of a series comes down to understanding the limiting behaviour of the sequence of partial sums. So, instead of asking about the series, let's step back for a second and ask how the sequence of partial sums might behave. If the series diverges, there are basically three possibilities:

1. It could be that for any $M > 0$, there exists an $N$ large enough that $n > N$ implies that $S_n > M$. In this case, we say that the sequence of partial sums (and the corresponding series) diverge to $\infty$. For example: $$ \sum_{n=1}^{\infty} 1 \qquad\text{or}\qquad \sum_{n=1}^{\infty} \frac{1}{n}. $$
2. It could be that for any $M > 0$, there exists an $N$ large enough that $n > N$ implies that $S_n < -M$. In this case, we say that the sequence of partial sums (and the corresponding series) diverge to $-\infty$. $$ \sum_{n=1}^{\infty} \frac{1}{\log(\frac{1}{n})}. $$
3. It could be that neither of these things occurs. In this case, the sequence (and the corresponding series) simply diverge, and no other descriptors are necessary, though we might be able to describe the behaviour in a little more detail if we care to. For example, the partial sums could bounce back and forth between to specific values, or they could oscillate with increasingly large oscillations, or they fall into some cycle, or they could simply bounce around at random.

In the case of the series presented in the question, the alternating series test shows that the series diverges (as you point out), but this is about the most that we can say without using some other tools. Very roughly speaking, we can see that the difference between consecutive partial sums increases to 1, so the sizes of the oscillations are bounded. There is also enough symmetry in the series to imply that we are not consistently adding more than we are subtracting, hence it is reasonable to think that the sequence of partial sums oscillates, but is bounded.