New Notes Blog: `a_n = (-1)^n(n/(n+1))` Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges, find its...

Friday, 19 December 2014

$\displaystyle{a}_{{n}}={{\left(-{1}\right)}}^{{n}}{\left(\frac{{n}}{{{n}+{1}}}\right)}$ Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges, find its...

A sequence of real numbers $\displaystyle{\left({a}_{{n}}\right)}$ is said to be convergent if $\displaystyle\forall\epsilon>{0},$ there $\displaystyle\exists{N}\in\mathbb{N}$ such that $\displaystyle\forall{n}>{N}$ then $\displaystyle{\left|{a}_{{n}}-{a}\right|}<\epsilon.$

If the sequence is convergent, then $\displaystyle{a}$ is called the limit of the sequence.

In other words the sequence is convergent if the terms tend to a single value a $\displaystyle{n}$ increases to infinity. That single value is called the limit of the sequence.

Let us first calculate limit of sequence with $\displaystyle{n}$ th term $\displaystyle\frac{{n}}{{{n}+{1}}}.$

$\displaystyle\lim_{{{n}\to\infty}}\frac{{n}}{{{n}+{1}}}=$

Divide...

If the sequence is convergent, then $\displaystyle{a}$ is called the limit of the sequence.

In other words the sequence is convergent if the terms tend to a single value a $\displaystyle{n}$ increases to infinity. That single value is called the limit of the sequence.

Let us first calculate limit of sequence with $\displaystyle{n}$ th term $\displaystyle\frac{{n}}{{{n}+{1}}}.$

$\displaystyle\lim_{{{n}\to\infty}}\frac{{n}}{{{n}+{1}}}=$

Divide both numerator and the denominator by $\displaystyle{n}.$

$\displaystyle\lim_{{{n}\to\infty}}\frac{{\frac{{n}}{{n}}}}{{\frac{{n}}{{n}}+\frac{{1}}{{n}}}}=\lim_{{{n}\to\infty}}\frac{{1}}{{{1}+\frac{{1}}{{n}}}}=$

Since $\displaystyle\lim_{{{n}\to\infty}}\frac{{1}}{{n}}={0}$ we have

$\displaystyle\lim_{{{n}\to\infty}}\frac{{1}}{{{1}+\frac{{1}}{{n}}}}=\frac{{1}}{{1}}={1}$

This part of the sequence converges to 1 however, $\displaystyle{{\left(-{1}\right)}}^{{n}}$ has two distinct values $\displaystyle-{1}$ for odd $\displaystyle{n}$ and $\displaystyle{1}$ for even $\displaystyle{n}$ (these types of sequences are called alternating sequences). Therefore, the sequence will have two distinct accumulation points $\displaystyle{1}$ and $\displaystyle-{1}.$ Therefore, if we choose $\displaystyle\epsilon<{2}$ and either of the two points e.g. $\displaystyle{1}$ , we can always find some term for which $\displaystyle{\left|{1}-{a}_{{n}}\right|}>{2}$ no matter how big the $\displaystyle{N}$ we choose.

Therefore, we conclude that the sequence is divergent.

The image below shows first 50 terms of the sequence. We can clearly see the two accumulation points $\displaystyle{1}$ and $\displaystyle-{1}.$

New Notes Blog

Friday, 19 December 2014

$\displaystyle{a}_{{n}}={{\left(-{1}\right)}}^{{n}}{\left(\frac{{n}}{{{n}+{1}}}\right)}$ Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges, find its...

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Friday, 19 December 2014

Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges, find its...

No comments:

Post a Comment

In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle{a}_{{n}}={{\left(-{1}\right)}}^{{n}}{\left(\frac{{n}}{{{n}+{1}}}\right)}$ Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges, find its...

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?