Real Sequence/Examples/n over (n+1)

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.

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.

$\blacksquare$

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 $1.2.1 \, \text {(a)}$