Divergence and Convergence for Series

Question

My textbook states that: If $\lim\limits_{x \to \infty } a_n$ doesn't exist OR $\lim\limits_{x \to \infty } a_n$ $\ne$ 0, then $$\sum_{n=1}^\infty a_n$$ would be divergent. My question is: Why are the two conditions stated with "Or". Say if my limit is 1, 1 $\ne$ 0 but it is existent. Is there something I'm missing or am I just lacking some brain cells XD

**Note: I get it now. Turns out I lack brain cells in the morning. Thanks for answering my question **

** Edit: Turns out I can't spell either **

Answer 1

$\lim_{n\rightarrow\infty}a_{n}\ne 0$ means that the limit of $(a_{n})$ exists, and the limit is not zero.

A sequence cannot both exist and not exist, so that is the reason putting the word OR.

$\lim_{n\rightarrow\infty}(-1)^{n}$ does not exist, and $\displaystyle\sum_{n=1}^{\infty}(-1)^{n}$ is divergent.

One cannot say $\lim_{n\rightarrow\infty}(-1)^{n}\ne 0$.

Answer 2

"OR" means that only one of the conditions needs to be satisfied for the series to diverge.

1. If the limit of the sequence doesn't exist (for example, the sequence alternates between $1$ and $-1$), the series diverges.

2. If the limit of the sequence exists but is not zero (for example, it is $1$), then the series also diverges.

Answer 3

We have two possibilities

• limit doesn’t exist

or

• limit exists and $\neq 0$

there is no contradiction in the statement.

Answer 4

A requirement for a convergent series is that:

$$\lim_{n\to\infty} a_n = 0$$

This is known as the $n$th term test. For a series to be convergent, it must satisfy the baseline that the sequence approaches $0$ as $n$ approaches $\infty$. It does not necessarily mean that if the limit is $0$, then the series must converge, though.

The other way of saying this is that a series is divergent if its sequence $a_n$ does not approach $0$ as $n\to\infty$:

$$\lim_{n\to\infty} a_n \neq 0$$

And for 'does not exist':

$$\lim_{n\to\infty} (-1)^n = \text{DNE}$$

If the limit is not $0$ it is divergent.

Answer 5

If the limit of your sequence is 1 then the serie does not converge.

Your $a_n$ will be as close to 1 as you want after a given index N (definition of convergence to 1). That will correspond to add an infinite number of ones's which is infinite (the serie diverge)