Convergent Real Sequence/Examples/x + x^n over 1 + x^n
Jump to navigation
Jump to search
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:
- $\ds \lim_{n \mathop \to \infty} \dfrac {x + x^n} {1 + x^n} = \begin {cases} 1 & : x \ge 1 \\ x & : -1 < x < 1 \\ 1 & : x < -1 \\ \text {undefined} & : x = -1
\end {cases}$
Proof
Let $\size x < 1$.
We have:
| \(\ds \dfrac {x + x^n} {1 + x^n}\) | \(\to\) | \(\ds \dfrac {x + 0} {1 + 0}\) | \(\ds \text {as $n \to \infty$}\) | Sequence of Powers of Number less than One | ||||||||||
| \(\ds \) | \(=\) | \(\ds x\) |
Now let $\size x > 1$.
We have:
| \(\ds \dfrac {x + x^n} {1 + x^n}\) | \(=\) | \(\ds \dfrac {\dfrac 1 {x^{n - 1} } + 1} {\dfrac 1 {x^n} + 1}\) | dividing top and bottom by $x_n$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {0 + 1} {0 + 1}\) | \(\ds \text {as $n \to \infty$}\) | Sequence of Powers of Reciprocals is Null Sequence | ||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
Then when $x = 1$:
| \(\ds \dfrac {x + x^n} {1 + x^n}\) | \(=\) | \(\ds \dfrac {1 + 1^n} {1 + 1^n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {1 + 1} {1 + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
When $x = -1$:
| \(\ds \dfrac {x + x^n} {1 + x^n}\) | \(=\) | \(\ds \dfrac {\paren {-1} + \paren {-1}^n} {1 + \paren {-1}^n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {-1 + 1} {1 + 1}\) | for even $n$ | |||||||||||
| \(\ds \) | \(\) | \(\ds \dfrac {-1 + -1} {1 + -1}\) | for odd $n$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) | for even $n$ | |||||||||||
| \(\ds \) | \(\) | \(\ds \text {undefined}\) | for odd $n$ |
Hence the result.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.20 \ (2)$