Alternating series test Puns

I am planning on take tonight off to study up for my worst module so far in my class. I don't have much time either, and when I google videos I see hour long explanations for each thing I listed in my title. I genuinely don't have a good 3-4 hours to set aside for this, so I was hoping someone who knows better could recommend me a shorter video(s) for the said topics with infinite series. Would make my night tonight when I come home! Cheers!

Also put etc. because I don't know what general subtopic that is of infinite series, but in my module it says "comparing"

Alternating series test:

1. All Un's must be positive
2. The limit of Un as n approaches infinity must be 0
3. The series is eventually non-increasing

https://i.imgur.com/Xmr19Sa.png

I thought for sure that the answer was C. The equation I figured out was (1/3^n - 1/2^n) Plugging in some terms into the calculator shows 2 things, that it approaches 0, and that every term is negative (which violates our rule that all Un's have to be positive, which is why I thought C was the answer). So why is the answer A?

I know it can't be D because that's what we're solving for

It can't be B because that's true

I thought it couldn't be A because the terms do keep increasing, they keep heading towards zero as you substitute in bigger and bigger numbers for n.

I also dont understand how to find the sum of an alternating series. We have a formula for a geometric series and for a finite arithmetic series, and we also know how to develop a formula when the terms cancel out, but none of those apply here. It says to use Reimann's rearrangement theorem, but that only gives me the ratio to the original series rather than the sum, right?

To preface: The Alternating Series Test has two "steps" to prove it is convergent.

Take the series Σ (n = 0; ∞) of (-1)^n * bn

In order to be convergent:

1. lim (n -> ∞) bn = 0

2. bn must be constantly decreasing

I am trying to find shortcuts for the Power Series exam I have coming up, and I'm having a difficult time thinking about a function where the lim −> ∞ = 0 but isn't constantly decreasing.

Are there major cases I will have to watch out for in which the first condition is met, but the second is not?

Thanks.

I havent seen anyone who got it

We’re learning alternating series test and the whole class is confused on why the series needs to be decreasing to pass the test. I get that it’s a part of the proof, but if the series goes to 0 as n tends to infinity, then it has to decrease. Why does the decreasing thing need to be a separate condition?

Edit: rephrase question

I feel as though I'm developing a solid understanding of the proof! This is exciting! However, there is one part that I can not find a proof for online.

The corollary used at 2:05 in the following video : https://www.youtube.com/watch?v=OcM3tGvLFEI&t=634s.

I understand it's necessary if one wants to declare that the series used for the proof converges, however I am unsure where it comes from, or where it's proved. Any help would be appreciated. Thanks!

The alternating series test is as follows. (Sorry for bad formatting).

I don't understand why the second condition (2.) needs to be in place, as in why the function needs to be decreasing, or that it needs to be decreasing beyond some integer N. Can someone give me an example of a function a sub n that satisfies all the other requirements (>0, limit to infinity goes to 0) except the second condition, where the alternating series with a sub n would diverge?

Thanks!

https://preview.redd.it/u4ervyililn21.png?width=957&format=png&auto=webp&s=64dace678b3fced50c50265c02550dab56126336

The series is: [Infinite series],n=1 (-2)^n * (x^n ) / (n^1/4 )

The problem asks for interval of convergence and radius of convergence;

So far I have:

Interval (-1/2<x<1/2)

f(-1/2)=[Infinite series],n=1 1/n^1/4 => P<1 => Diverges

Here is where my issue lies:

f(1/2)=[Infinite series],n=1 (-1)^n /n^1/4

The alternating series test fails, as well as every other test I have used so far. What test or property am i needing to use here to determine f(1/2)'s convergence or divergence?

https://i.imgur.com/sL8MFkz.png

Why must b be positive? Wouldn't the alternating negative sign still cause the entire thing to alternate, just in a different order i.e.: (+)(-)(+)(-) instead of (-)(+)(-)(+)

Or, to even further the idea of what I'm saying, couldn't a negative sign on b sub n just be factored out of the sum?

going over lecture notes and the alternating series test has two conditions and one is lim k→∞ |ak| = 0 and there is no written explanation of what this means so i was wondering if someone could briefly explain what is means?

ive looked online and their doesn't seem to be any simple explanations

Still starting out so it's not like I'm against crazy views but I've been doing a timberborn series and every other video I'll post something other just to try things out. The other videos don't get a ton of views so I'm wondering if that hurts the algorithm's perception of my videos overall?

This is the second time that I encountered a problem where this happened. The first time, I chalked it up as me probably screwing something up in the alternating series test, but now I'm not sure.

Since if the Root test converges, it means the series should be absolutely convergent. However, how is that possible if the alternating series test says that it diverges?

Here's the problem if you want to see it : https://puu.sh/s11Mn/92c1169071.png

As aforementioned, I first used the alternating series test and then the root test where I got differing results.

The reason why I believe it fails the Alternating series test is because if you follow natural law rules, you get ln((2n+1)/(n+3))^n which you can take the limit inside of and get ln(2) which is not equal to 0 ( which fails the ALS test because limit doesn't equal 0)

[Real Analysis] Alternating Series Test, Showing |u(n+1)| <= |un|

I'm proving a series is convergent using the alternating series test. I am able to show un -> 0 as n-> infinity, but I am unable to show the other part of the test |u(n+1)| <= |un|

The series is: (1-log2)-(1-2log(3/2)+(1-3log(4/3)-...+

I have figured un = (-1)^(n+1) (1-nlog((n+1)/n))

I've tried various properties of logarithms, but I can't get anything that definitively shows |u(n+1)| <= |u(n)|. I do know that the series is convergent, so not being able to show this is not a reflection of this series perhaps not being convergent.

Any hints in the right direction would be welcome. Thanks!

We're studying infinite series in my BC Calc class right now, and we just went over the alternating series test. One of the requirements is that the function must be decreasing as n approaches infinity, and so our teacher taught us that, in cases where it isn't obvious, we should use the first derivative test to find if the function is decreasing over time.

However, one of the other requirements of the test is that the limit as n approaches infinity must equal 0. I reasoned that if the infinite limit is 0, and the function is always positive, that it would have to decrease as n approaches infinity. I brought this up to my teacher, and he said to use the first derivative test just to make sure. Can anyone here find an exception to the "rule" I came up with?

http://imgur.com/Zaub8ga

I know that this series converges, but I'm have trouble proving that the series is decreasing (finding a term N, where a N+1 < a N) Also I need to show absolute convergence or conditional convergence and I am unsure how to show this. Thanks.

Ok, so I have an exam in 2 days. I'm more than halfway done with this chapter. I'm doing fairly well. I'm posting to see if there are other ways you recognize when to do each of the tests; such as the basic comparison test, the root test, the ratio test, the alternating series test, limit comparison test, and the alternating series test. I figure we all think differently, and I wanted to look at it this from every perspective.

I'm not that great with the alternating series test yet, but luckily I have my exam Wednesday night (it's still Monday for me). I'm about to finish all of my homework tonight. Tomorrow after work, I'll be at school in the math lab doing extra problems. I just wanted to get some input to refine these concepts. Thank you!

https://gyazo.com/aae9a63200a0ebdd32d8cb836a052b78

When I do the alternating series test, it must pass two conditions to be convergent.

1. Must pass divergence test.
2. a_n+1 must be decreasing; a_n+1 <= a_n.

Lets say one of the problems I do fails the divergence test, that means it automatically diverges right?

Should I still see if it satisfies condition two, or does that mean the test is inconclusive?

Thanks.

Edit: a_ -> a_n