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

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