The Convergent Sequences

I started treating the topics of sequences yesterday and i was able to look into the
common types of sequences we have.But today i will be looking at a very important aspect
of the sequences which is its ability to converge.

Literally when we hear the word "converge",we know it is the ability of something to
approach or finally assemble at a particular place.A sequence has this ability,for
example the sequence \begin{equation}1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},...\end{equation}
is getting closer and closer to the number 0.hence we say that this sequence converges to
0 or the limit of the sequence is the number 0.

Now lets take a proper look at the definition of a sequence using the mathematical
approach.
"A sequence $\{s_n\}$ converges to a number $L$ if the terms of the sequence get closer
and closer to $L$."
Now this is how the concept of convergency is viewed with respect to mathematics.
But there are some case of sequences that might not satisfy our definition for example,
\begin{equation}0.1, 0.01, 0.02, 0.001, 0.002, 0.0001, 0.0002, 0.00001, 0.00002, . . .\end{equation} 
Surely this should converge to 0 but the terms do not get steadily "closer and closer"
but back off a bit at
each second step.
Also, the sequence \begin{equation}
0.1, 0.11, 0.111, 0.1111, 0.11111, 0.111111, . . .
\end{equation}
is getting "closer and closer" to 0.2, but we would not say the sequence converges to 0.2 .
with these few illustrations, i believe we should have a clear picture of what a
convergent sequence is all about.
We will now move into a more profound definition of a sequence using the concept of
"limits".

Limits of a Sequence: Let $s_n$ be a sequence of real numbers, then $s_n$ is said to
converge to a number $L$ denoted as $\lim_{n\rightarrow\infty}=L$ or
$s_n\rightarrow{L}$ as $n\rightarrow\infty$ provided that for every $\epsilon>0$ there
exists an integer $N$ such that $|s_n-L|<\epsilon$ whenever $n\geq{N}$
Now we must know that a sequence that converges is called a convergent sequence while a
sequence that fails to converge is called a divergent sequence.

Example: Check if \begin{equation}
\frac{n^2}{2n^2+1}=\frac{1}{2}
\end{equation} it is convergent or not.

Solution: \begin{equation}
\frac{n^2}{2n^2+1}=\frac{1}{2}
\end{equation} by interpretation it means
\begin{equation}
\{\frac{n^2}{2n^2+1}\}^{\infty}_1
\end{equation}
divide the L.H.S by the highest power which is $n^2$
\begin{equation}
\lim_{n\rightarrow\infty}\frac{n^2}{2n^2+1}=\frac{1}{2}
\end{equation}
\begin{equation}
=\lim_{n\rightarrow\infty}\frac{1}{2+\frac{1}{n^2}}
\end{equation}
\begin{equation}
=\frac{1}{2+\lim_{n\rightarrow\infty}(\frac{1}{n^2})}=\frac{1}{2}
\end{equation}