# Nature of the sequence

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• April 16th 2013, 05:48 PM
calmo11
Nature of the sequence
I have already posted this question but I accidentally asked for the nature of the 'series', where as I wanted someone to help me with,
how I can tell if the following sequences {an} diverge or converge? And how to justify it?
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• April 16th 2013, 07:30 PM
chiro
Re: Nature of the sequence
Hey calmo11.

The first thing you should do is list the type of series (alternating, non-alternating in sign) and list the different kinds of tests for that particular series. Can you do this to start off with?
• April 16th 2013, 07:35 PM
Soroban
Re: Nature of the sequence
Hello, calmo11!

Quote:

How I can tell if the following sequences diverge or converge? And how to justify it?
Take the limit of $a_n$ as $n\to\infty.$
If the limit is finite, the sequence converges.

Quote:

$(a)\;a_n \:=\:\frac{(\text{-}1)^n}{n}$

We see that: . $\displaystyle \lim_{n\to\infty}\frac{(\text{-}1)^n}{n} \;=\;0$

The sequence converges.

Quote:

$(b)\;a_n \:=\:\frac{(\text{-}1)^n + n}{(\text{-}1)^n - n}$

Divide numerator and denominator by $n\!:\;\;\frac{\dfrac{(\text{-}1)^n}{n} + 1}{\dfrac{(\text{-}1)^n}{n} - 1}$

Then: . $\lim_{n\to\infty}\frac{\dfrac{(\text{-}1)^n}{n} + 1}{\dfrac{(\text{-}1)^n}{n} - 1} \;=\;\frac{0+1}{0-1} \;=\;-1$

The sequence converges.

Quote:

$(c)\;a_n \:=\:\sqrt{n+1} - \sqrt{n}$

Multiply by $\tfrac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1}+\sqrt{n}}\!:\;\;$

. . $\frac{\sqrt{n+1} - \sqrt{n}}{1}\cdot\frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}} \;=\; \frac{(n+1) - n}{\sqrt{n+1} + \sqrt{n}} \;=\;\frac{1}{\sqrt{n+1} + \sqrt{n}}$

Then: . $\lim_{n\to\infty}\frac{1}{\sqrt{n+1}+\sqrt{n}} \;=\;\frac{1}{\infty} \:=\:0$

The sequence converges.