Whats the difference between a series and sequence?
I was looking at a question earlier that involved sequences and found out that the sequence converged to 0 but the series diverged to infinity. How is that possible? for example the sequence was $a_n$ $=$ $\frac{1}{x_n}$ The question was here: Convergence of $x_n=f\left(\frac{1}{n}\right), n\geq 1$
A sequence is an infinite list of numbers: $$a_1,a_2,a_3,\ldots$$ while a series is the formal expression you get when you insert "plus" symbols: $$a_1+a_2+a_3+\ldots$$ Loosely stated, the sequence converges iff the numbers in the list approaches some limit number $a$. And the series converges iff some number $s$ can be said to be the sum of the series.
If the series converges to some sum $s$, it is not hard to show that the corresponding sequence converges to $a=0$. However, the converse is not true; if the sequence converges to $a=0$, the series obtained by inserting plusses may not converge. In a sense this happens when the terms $a_n$ tend to zero too slowly.
A series is in some sense a type of sequence. However, it is a sequence of partial sums.
For example, if we take some sequence $\{a_n\}_{n\geq 1}$, then we can in turn retrieve a series from this sequence by considering the following partial sums:
$S_N=a_1+...+a_n=\sum_{n=1}^{N}a_n$
Then if we consider the following sequence: $\{S_N\}_{n \geq 1}$ We have actually defined a series!
$S_1=a_1$
$S_2=a_1+a_2$
and so on.
Thus, the difference is the following:
Consider the sequence $\{a_n\}_{n \geq 1}$ defined by $a_n=\frac{1}{n}$.
It is intuitively clear (and if not, use the Archimedean property) that $\lim_{n \to \infty} \frac{1}{n}=0$.
Now, we can consider the sequence of partial sums:
Let $\{S_n\}_{n \geq 1}$ be defined by $S_n=1+\frac{1}{2}+...+\frac{1}{n}$
Then $$\lim_{n \to \infty} S_n=\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{k}=\sum_{k=1}^{\infty}\frac{1}{n}$$
Which is called the harmonic series, and it diverges.
We can show this by either the integral test, or just note:
$\begin{align} &1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+...\\ >&1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+...\\ =&1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+... \end{align}$
Which very clearly diverges.
As an aside, it should be very clear that $\{S_n\}$ will not converge at all if $\{a_n\}$ does not, but the converse is not true.
If you want an example where they both converge, just take $\{a_n\}$ to be defined by $a_n=\frac{1}{2^n}$. Then you can consider the sequence of partial sums for $a_n$ (and hence, define a series.)
Then $\{a_n\} \to 0$ as $n \to \infty$. Yet we also have that
$$\lim_{n \to \infty} S_n=\lim_{n \to \infty} \sum_{k=1}^{n}\frac{1}{2^k}=\sum_{k=1}^{\infty}\frac{1}{2^k}=\frac{1/2}{1-1/2}=1$$
To see a derivation of the penultimate equality, you can look further into what are called geometric series
An essential difference is that
a sequence has one specific meaning and
a series has two different specific meanings.
For convenience only we look at real-valued sequences and series.
Sequences: A sequence of real numbers \begin{align*} \left(a_n\right)_{n\geq 0}=(a_0,a_1,a_2,\ldots) \end{align*}
is a function from $\mathbb{N}$ to $\mathbb{R}$. The index set $\mathbb{N}$ is the domain of the function and the real numbers $\mathbb{R}$ is the codomain.
Now let's have a look at series. We have to clearly state the meaning(s) of the symbol \begin{align*} \sum_{n=0}^{\infty}a_n\tag{1} \end{align*} with $a_n$ being real numbers.
Series (as sequences): Infinite series are sequences of partial sums
When looking at a sequence $(a_0,a_{1},a_{2},\ldots)=(a_n)_{n\geq 0}$ we consider \begin{align*} (a_0,a_{0}+a_{1},a_{0}+a_{1}+a_{2},\ldots) = \left(\sum_{n=0}^{N}a_n\right)_{N\geq 0} \end{align*} and define the infinite series (1) as the sequence of partial sums \begin{align*} \sum_{n=0}^{\infty}a_n:= \left(\sum_{n=0}^{N}a_n\right)_{N\geq 0}\tag{2} \end{align*}
So, the LHS of (2) is just a new and very convenient representation for the sequence of partial sums \begin{align*} \left(s_N\right)_{N\geq 0}, \qquad s_N=\sum_{n=0}^{N}a_n\tag{3} \end{align*}
The symbol $\sum_{n=0}^{\infty}a_n$ has also a second meaning, namely
Series (as values): Infinite series are the limits of the sequences of partial sums
An infinite series $\sum_{n=0}^{\infty}a_n$ is said to be convergent, definitely divergent or indefinitely divergent according as the sequence of its partial sums shows the behaviour indicated by those names.
If, in the case of convergence, the sequence of partial sums, $s_N\rightarrow s$, then we say that $s$ is the value or the sum of the convergent infinite series and we write for brevity
\begin{align*} \sum_{n=0}^{\infty}a_n:=\lim_{N\rightarrow\infty}s_N=s \end{align*} so that $\sum_{n=0}^{\infty}a_n$ denotes not only the sequence $(s_N)_{N\geq 0}$ but also the limit $s$, provided this limit exists.