Is the Sequence a_n=(n-1)/n Monotonically Increasing and Convergent?

[kreil]

determine the monotonicity of [itex]a_n=\frac{n-1}{n}[/tex]<br /> <br /> heres my work...<br /> <br /> $$\frac{a_{n+1}}{a_n}=\frac{ \frac{n}{n+1}}{\frac{n-1}{n}}$$<br /> <br /> $$=\frac{n^2}{n^2-1}>1$$<br /> <br /> Therefore the sequence is monotone increasing.<br /> <br /> But...when you look at the original expression for a_n, it looks like it is always LESS than one...does this impact anything at all?<br /> <br /> <br /> Josh[/itex]

 

[Tide]

The sequence approaches the limit 1 from below so it's clearly monotonically increasing.

 

[quasar987]

A sequence can be strictly increasing but never greater than a certain number. This is what limits are all about.

 

[HallsofIvy]

It has an important impact! One of the fundamental properties of the real numbers is the "Monotone Convergence Property". If an increasing sequence of real numbers has an upper bound, then it converges.

What you have here is an increasing sequence that has every number larger than or equal to 1 as an upper bound. 1 is its "least upper bound" and so the sequence converges to 1.