Thread: A little infinite series help

Note: I'm not sure how to make everything look all nice, so I'm putting everything in words.

Well, I'm a little fuzzy on my infinite series, so here we go.

Firstly, if the terms of a series approach 0, then the the series may or may not be convergent.
If it does not approach 0, then is definately does not converge.
Is this correct?

So, I have an infinite series where a sub n [the function after the E thing. I'm not sure what the accepted notation is around here] is (1/(ln(n)). This approaches 0. Is it convergent? How do I tell?

Thanks in advance.

Originally Posted by Nerd

Firstly, if the terms of a series approach 0, then the the series may or may not be convergent.
If it does not approach 0, then is definately does not converge.
Is this correct?

you are correct!

E (1/(ln(n)).

what tests have you learnt so far. let's try the comparison test, i think that's the easiest way to do this.

note that 1/n < 1/ln(n) .........since n is larger than ln(n) for all n

so E 1/n < E 1/ln(n)

now since E 1/n diverges, E 1/ln(n) diverges by the comparison test

I will write a sub n as An and b sub n as Bn

The comparison test says:

Let E An and E Bn be series of positive terms

(1) If E Bn converges and An <or= Bn for all n, then E An also converges
(2) If E Bn is divergent and An >= Bn for all n, then E An also diverges

Ah. Can't believe I didn't think of that. Thanks.

Any others ways to show this is convergent? I know a few more tests...I just forgot what they were. Oh, and can someone explain what the integral test is?

Originally Posted by Nerd

Ah. Can't believe I didn't think of that. Thanks.

Any others ways to show this is convergent? I know a few more tests...I just forgot what they were. Oh, and can someone explain what the integral test is?

we can do this by the integral test also.

the integral test only works if An is nonincreasing , and all the terms are nonnegative (very important, the test is invalid if these conditions don't hold)

this problem satisfies the conditions for n>1

so according to the integral test, a series E An converges if int{1:infiniyt} (f(x)) dx < + infinity (that is, it is a finite number) and E An diverges if int{1:infiniyt} (f(x)) dx = + infinity

note: f(x) is the analogous function to An, in this case it would mean f(x) = 1/ln(x)

do you want to see it done?

Originally Posted by Jhevon

do you want to see it done?

nevermind, that integral's a pain.

comparison test is by far the easiest way to go

I remember that integral really looking hard. I just wanted to know the test, so thanks. It was a little unclear in my notes.

Originally Posted by Nerd

I remember that integral really looking hard. I just wanted to know the test, so thanks. It was a little unclear in my notes.

sure thing. other tests you might want to brush up on

You should know these:
root test
ratio test
alternating series test
geometric series
p-series test
test for divergence (the one you mentioned at the beginnig)

You might consider learning these:
limit comparison test
telescoping

Originally Posted by Jhevon

sure thing. other tests you might want to brush up on

You should know these:
root test

The root canal test is not usually taught in a standard Calculus course. It is omitted.

Originally Posted by ThePerfectHacker

The root canal test is not usually taught in a standard Calculus course. It is omitted.

really? i've seen it in every math class i've done that has series in it. calc 2, calc 3 and the course i'm taking now

lol, root canal test. it's not that painful if you apply it to the right problem

I must admit that I've never learned the root canal test. I did learn the root test.

root test - know it
ratio test - know it
alternating series test - know it
geometric series - I forget on occasion, but generally know it
p-series test - my teacher would kill me if I didn't know p-series

You might consider learning these:
limit comparison test
telescoping

No clue. Maybe we'll get to it in class

And that reminds me. In my physics class, we were exchanging math jokes and pick up lines. Is this a healthy social life? I can only remember about one or two though (I can't remember what I ate yesterday either, but I'm almost getting an A in Calc...).

Originally Posted by Nerd

I must admit that I've never learned the root canal test. I did learn the root test.

root test - know it
ratio test - know it
alternating series test - know it
geometric series - I forget on occasion, but generally know it
p-series test - my teacher would kill me if I didn't know p-series

No clue. Maybe we'll get to it in class

And that reminds me. In my physics class, we were exchanging math jokes and pick up lines. Is this a healthy social life? I can only remember about one or two though (I can't remember what I ate yesterday either, but I'm almost getting an A in Calc...).

i think when TPH said root canal test, he was making a joke, i think he was implying that the root test is as painful as a root canal (i don't know why, he's great at it).

like i said, telescopic and limit comparison tests are optional, maybe you'll see them, maybe you won't, i don't see problems that require them very often

let's put it this way, i don't think exchanging math jokes and pick up lines make your social life unhealthy

you can post some that you remember in chat, or there's a math jokes thread somewhere, post them there

here's the thread http://www.mathhelpforum.com/math-he...them-here.html