Real Sequence/Examples/n over (n+1)
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Examples of Real Sequence
The real sequence $S$ whose first few terms are:
- $\dfrac 1 2, \dfrac 2 3, \dfrac 3 4, \dotsc$
can be defined by the formula:
- $S = \sequence {\dfrac n {n + 1} }_{n \mathop \ge 1}$
$S$ is strictly increasing.
Proof
Let $s_n$ denote the $n$th term of $S$.
We have:
\(\ds s_{n + 1} - s_n\) | \(=\) | \(\ds \dfrac {n + 1} {n + 2} - \dfrac n {n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 - \dfrac 1 {n + 2} } - \paren {1 - \dfrac 1 {n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {n + 1} - \dfrac 1 {n + 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {n + 2} - \paren {n + 1} } {\paren {n + 1} \paren {n + 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {\paren {n + 1} \paren {n + 2} }\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds 0\) |
Hence $S$ is increasing by definition.
$\blacksquare$
Sources
- 1919: Horace Lamb: An Elementary Course of Infinitesimal Calculus (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Continuity: $2$. Upper or Lower Limit of a Sequence
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Example $1.2.1 \, \text {(a)}$