Integral Test for Series: Why It Works
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- Опубліковано 14 лис 2012
- Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) / patrickjmt !! Integral Test for Series: Why It Works.
In this video, I give a little geometric argument about the integral test for series and why it works when we compare our series to an improper integral.
absolutely amazing!
correction: monotone functions*. simply bounded or everywhere defined isn't enough. take for example the series: 1/n^2 + (sin(n*pi))^2. At every natural number n, (sin(n*pi))^2 = 0, so the series converges., however the integral clearly diverges since (sin(n*pi))^2 does not approach zero as a limit.
never heard of that, i will check it out
Clear as mud.
If he would simply position the mic so he isn't spitting into it, it would help. Usually, the mic is supposed to be positioned to the side of the mouth (depends on the mic).
Awesome, thanks
I wonder about what restrictions there are for when you are allowed to use this test. The series 1/(n-0.5)^2 will converge [by limit comparison test], but the integral will diverge since as x approaches 0.5, it shoots off towards infinity, p test: p=2. Clearly it should be true for monotone decreasing sequences, and by symmetry for monotone increasing sequences., but it should be occasionally true for more complicated non-monotone sequences. Perhaps if the function must be bounded
this was super helpful! finally someone who explained why you use the left or right version of the sum representation
I love youuuuu!!! Thank you so much!
You have no idea how "right on time" this video is. My whole class loves you!
Quick question for you Patrick. How do you film your film? Is there a webcam over your shoulder? I only ask as i have a coursework where you have to make a video of you explaining a concept, and I really like your style! Thanks man!
Awesome!
I always asked it myself.
i have a huge playlist for it
awesome timing, i was just preparing for seq and ser. you should have separate playlist for it. thanks
@ freelapdane. 1/x is the harmonic series and its exponent is equal to one. 1/x^2 is a p-series, in which the exponent is greater than one, so it will converge.
It's not about the sound quality. It's the popping that occurs when pronouncing letters like p (caused by strong bursts of air). A pop filter (basically a small sheet of some fabric I can't remember) helps stopping these bursts and thus eliminates most of the pops.
Patric JMT does it again! Your videos are so brilliant, so nail-biting I think I may be developing a nonfunctional habit from watching your videos, with all the nail-biting. I may be developing TMJ! Now I can go around telling people I got TMJ from watching Patric JMT. They won't know what I'm talking abut but that's okay, I'm used to that.
IF only you posted this a few weeks into the semester and I would have been golden!
You are in for serious fun, man. These things can become a real pain real quick.
its awesome that he posted this today because i'm doing the same topic, preventing me from procrastinating lol
i position it off to the side. i am not eating it while i make the video...
Patrick would you download a video about the difference between balanced and unbalanced coin cause I can't know the difference between them?
All you need to do is not speak directly into the mic. If the microphone is close to your mouth but not in line with your breath, that should eliminate the pops. If using a headset move up or down or if a mic on a stand, move to the side but point at your mouth. Should help a lot.
so it's pretty similar to direct comparison of series?
I have a question. If we were given the integral of 1/(infinity squared) . Wouldn't it be infinite? Does it being finite, mean that x eventually = 0?
Yes, as you approach infinity the function approaches 0. But what we are measuring here is the area under the function. The shape that we are considering has a hight of one at the left side, a infinite base, and the hight decreases towards zero as x approaches infinity. That area doesn't necessarily have to be finite since the base is infinite.
I hope I interpreted your question right.
Quick question, I will be taking Elementary Linear Algebra next semester and I hear it's even more abstract and difficult than Calc 2. Have you covered topics pertaining this subject in your vast collection? I'm not sure where your brilliance has ended :)
What he really needs is a decent pop filter, not a new microphone (of course that might help, too). They only cost around 20$.
So what you are trying to say is.. "If you draw boxes larger, it turns out divergent, but if you draw boxes smaller, it becomes convergent for 1/n Series." Right?
I'm so pissed at math right now. Well, more at myself I guess. I took an exam yesterday and got a problem wrong (I realized after walking out the door) because of a really subtle interpretation error. I knew literally EVERYTHING too. That's friggin' math for you.
my ears must be messed up, it does not sound that bad to me!
can you show me the proof of derivations: x^n=x^n+1/n+1
If they gave you more time for these long, arithmetic-ridden exams, it would be a little more fair. I'm so OCD with math, it takes
me forever to do things because I recheck everything, even when I know I'm right.
i hated series, it was never made clear at all what the hell we were doing, it was just kind of thrown at us like spaghetti on a wall. After cal 2 i still dont have a strong understanding of these convergence tests
how come 1/x is infinite while 1/x^2 is finite? for me it seams like both are infinite..
thanks for a awesome web show :)
Doesn't really sound that bad to me either!
you couldnt show what we needed
don't you get tired of that sharpie smell?
series is never fun