Does this series converge or diverge ?

sid9221

I have tried all the tests I know, non-null, comparison, ratio, alternating series test. None have conclusive results. Any ideas on how to proceed.

jsewell94

When you have a [itex]-1^{n}[/itex] in an infinite series like this, you are going to have to use the alternate series test.

1) Prove that the limit to infinity of [itex]\frac{1}{\sqrt{N+3}}[/itex] goes to zero.

2) Show that [itex]\frac{1}{\sqrt{N+3}}[/itex] is a decreasing sequence.

If you can show those two things, then you know that the series is convergent :D

Ray Vickson

No, having [itex]-1^n[/itex] would give no useful information. However, having [itex] (-1)^n[/itex] would be helpful.

RGV

sharks

Take the modulus of the entire expression, and then you'll end up with [itex]\frac{1}{\sqrt{n+3}}[/itex]. Since the latter converges to zero, therefore, so does the original expression. The series converges.

Dick

No. The sequence [itex]\frac{1}{\sqrt{n+3}}[/itex] converges to zero. The series is the sum of all those terms. It diverges.

Dickfore

It diverges conditionally.

Bohrok

It diverges absolutely but converges conditionally.

sharks

OK, to make it less confusing, here's a suggested solution...

Using the comparison test:
[tex]u_n=\frac{1}{\sqrt{n+3}}[/tex]
For n large, [tex]v_n=\frac{1}{\sqrt n}[/tex]
Using the p-series test, [itex]v_n[/itex] diverges.
Therefore, since [itex]u_n[/itex] is less than [itex]v_n[/itex], [itex]u_n[/itex] also diverges.
Hence, its sequence converges but its series diverges.

Is this correct?

sid9221

This should not be so complicated, its a past exam question in the "easier" section worth 3 marks in a 2 hour exam.

I was thinking about using the fact that "An absolutely convergent Series Converges".

Hence I used the integral test on [tex]u_n=\frac{1}{\sqrt{n+3}}[/tex] which equals infinity meaning it diverges.

So that was a fail !

Now the only other test is the alternating series test, clearly 1/sqrt(x) is decresing as x->inf, and the limit goes to zero.

So by the alternating series test the series converges ?(Basially what the second guy said)

Any logical inconsistencies with that proof ?

Dick

That's exactly what you want for a proof.

Dickfore

Sorry, it converges conditionally.

sharks

Now, i'm confused. I don't think i completely understand the meaning of those words.

Dickfore

A series is said to coverge/diverge absolutely iff the series of absolute values:
[tex]
\sum_{n = 1}^{\infty}{\vert a_n \vert}
[/tex]
converges/diverges.

There is a Theorem that states that if a series converges absolutely, then it also converges ordinarily.

The converse, however, is not true. Some alternating series satisfy the criterion that:
[tex]
\sum_{n = 1}^{\infty}{a_n}
[/tex]
converges, but the series diverges absolutely. For these series, it is said that they converge conditionally.

HallsofIvy

Actually, I don't believe I have ever seen the phrase "diverge conditionally". A series [itex]\sum a_n[/itex] converges conditionally if it converges but [itex]\sum |a_n|[/itex] does not converge.

sid9221, you say in your first post that you tried the "alternating series" test and it did not work. Could you explain what you did? It looks to me like it works nicely.

sharks

Too much confusion. I think i'll just stick with what i know. No offence.

SammyS

You can also combine pairs of terms: n = 1 & 2, n = 3 & 4, n = 5 & 6, etc.

[itex]\displaystyle \sum_{n = 1}^{\infty}\frac{(-1)^n}{\sqrt{n+3\,}}
=\sum_{n = 2,4,\dots}^{\infty}\left(\frac{(-1)^n}{\sqrt{n+3\,}}+\frac{(-1)^{n-1}}{\sqrt{n-1+3\,}}\right)[/itex]

[itex]\displaystyle =\sum_{k = 1}^{\infty}
\left(\frac{1}{\sqrt{2k+3\,}}+\frac{-1}{\sqrt{2k+2\,}}\right)[/itex]

Combine the rational expressions into one.

Use comparison test & p-test.