Convergent Real Sequence/Examples/x + x^n over 1 + x^n

Example of Convergent Real Sequence
The sequence $\sequence {a_n}$ defined as:


 * $a_n = \dfrac {x + x^n} {1 + x^n}$

is convergent for $x \ne -1$.

Then:
 * $\displaystyle \lim_{n \mathop \to \infty} \dfrac {x + x^n} {1 + x^n} = \begin {cases} 1 & : x \ge 1 \\ x & : -1 < x < 1 \\ 1 & : x < -1 \\ \text {undefined} & : x = -1

\end {cases}$

Proof
Let $\size x < 1$.

We have:

Now let $\size x > 1$.

We have:

Then when $x = 1$:

When $x = -1$:

Hence the result.