What am I doing wrong? (Monotonicity of sequence)

[theneedtoknow]

Homework Statement

Determine the monotonicity of the sequence with An as indicated


An = (-1)^(2n+1) * n^0.5

The Attempt at a Solution

well without even doing any tests, I can see that 2n+1 is always odd, so the -1 will make every term negative
so it simplifies to
An = - root(n)
which is obviously a decreasing sequence...

Now here is where my problem somes in
If i do a ratio test of An+1 / An I get

An+1 / An = [ (-1)^(2(n+1)+1) * root (n+1) ] / [ (-1)^(2n+1) * root (n) ]
= [ (-1)^(2n + 3) * root (n+1) ] / [ (-1)^(2n+1) * root (n) ]


when i divide (-1)^(2n + 3) by [ (-1)^(2n+1) i get (-1) ^ 2 which is 1
so An+1 / An reduces to = root (n+1) / root (n) which is always greater than 1... now if its greater than one, doesn't it mean that each term is greater than th last, and the seqence is increasing? but from simplifying the expression for the term An I can clearly see that it hs to be decreasing...so where am I screwing up? Am i not supposed to apply ratio tests to sequences with each term being negative? or does the ratio test work the opposite way with sequences that are always negative (ie i need it to be less than one for it to be increasing and more than 1 for it to be decreasing?) This is probably a really dumb question.

 

[statdad]

Write out the first three or four terms of A_n explicitly to see what the terms
look like. That will show you whether the sequence is monotone increasing, decreasing, or neither.
Once you believe you know which it is, you need to prove it. Examine

<br /> A_{n+1} - A_n<br />

If this is positive you have one answer, if it is negative you have another.

Hint: what do you know about (-1)^{2n+1} ?

 

[theneedtoknow]

Well I understand the reasoning and i know what it tells me about the -1 (i mention it in the post) so I can see that the sequence is decreasing just from lookin at the simplified form . My question is more why the ratio test fails, whether the ratio test can only be applied to sequens with all positive terms, or whether i can be applied to negative sequences but in reverse (ie An+1/An < 1 means increasing)

 

[statdad]

Sorry - I simply missed part of your question.
You will find the Ratio Test stated in several cosmetically different, yet equivalent ways, but all of them deal with the problem of investigating convergence of a series, not a sequence. Further, when the ratio test is used, it is specified that for the series

<br /> \sum_{n=1}^\infty a_n<br />

the terms a_n are all positive , or the test is performed with the absolute values of the terms, as

<br /> \lim_{n \to \infty} \left\vert \frac{a_{n+1}}{a_n} \right\vert<br />

The ratio test doesn't apply to your question because there is no question of convergence, as you do not have a series: you wanted to know something about the monotonicity of the sequence.

Since the powers of -1 cancel out in the ratio of terms, your calculation showed that the magnitude of A_n increases as n increases. This, coupled with the fact that all terms are negative, gives the answer to the monotonicity question.

Sorry for the wordiness, and for missing part of your original question.

 

[theneedtoknow]

Thanks so much :) That makes sense