Sequences and series (part 2)

Finding the sum of an infinite geometric series.

[@jellie1443]

You thoroughly taught me Sequences and series in just 20 mins while I took 6 hours in Math Training and didn't fully get it.

[@mokshshah9305]

Same

[@khanacademy]

You're right. Thanks for noticing. Seems like my brain malfunctions once every 3 videos or so. I guess my mistakes keep people on their toes.

[@novaastronomia8720]

Hahaha, I knew it.

[@nineironshore]

pls don't do that on purpose

[@honeywhisker]

I find this guy's commentary so enthusiastic. Im here because my class is annoying (its full of class clowns) and I have a test tomorrow. It's easier to learn from here than trying to understand verbal concepts and straight-on paper

[@hmdlamin93]

dude, u simply dominate math!! you are the chuck norris of mathematics! :)

[@jarrasoma]

WOW! I am totally amaized with this infinity to finite number! Watching those videos is much more interesting than anything I ever watched. Makes me want to buy a pentablet for this dude. Hes doing amaizing work!!!! Thank you so much. Alot of good karma for you!!!!

[@fspopshovit]

It only applies to numbers between -1 and 1 (but not -1, 0 or 1). If you apply the infinite geometric sequence to a number higher than 1, your answer will just be infinity itself. Taking the infinite geometric sequence of a negative number below -1 (including -1) is undefined. Because -x^∞ has no answer. The obvious answer would be -∞ or +∞, but infinity itself is not even nor odd. Hope this helped and that last bit wasn't too confusing..

[@runningmarvel]

i like your teaching style, and the odd mistake is good, because it keeps people thinking. keep it up, i like it!

[@Ankur11ify]

Sal, you really need to put up videos on testing a series' convergence or divergence through Integral Test, Comparison Test, Ratio and Root Test.

[@CynicalApple]

that is the most mind-blow-iest thing I've heard all year.

[@tlsauer]

Isn't the first term in the series 1/2^K equal to 1 and not zero? Thanks for the video and thanks for all the answers to the SAT test problems!

[@TareqKhan0]

Yes, it should be 1.

[@imegatrone]

I Really Like The Video From Your Finding the sum of an infinite geometric series.

[@tska84]

lol i love how he always catches his mistakes and is like wtf was i thinking.

[@vashfish]

I got some intuition on that last example for you from working with binary. What you're basically doing with the sum of 2^N is building the binary number 1.1111111111..... The decimal equivalent would be 9.999999999999...., which we know approaches 10. 10 in binary is 2.

[@GenericCoder]

This video is mind blowing thanks alot :D.

[@kypronite]

great job sal,I really like the way you use the box "1" to cover the mistake you made throughout the video.

[@waterbot]

Sal is loving this stuff

[@scorpionboy3]

That makes sense, because when you´re mutliplying (1/2) times itself, your basicly dividing it by 2 every time. So if you do this infinite times, you would be dividing the original (1/2) infinite times, making it smaller and smaller ,which then goes to zero.

[@elimayfield8689]

6:15 Memorize your exponent tables? I call it a calculator

[@sean9878]

Shouldn't it be a^n-1 then a^n??? Or a^n-2, a^n-1, a^n?

[@turbold1]

great video thnx sal hope to see more interesting videos! keep it up

[@professorlessor]

it may feel odd at first but it is a fundamental law of algebra... if a=b then multiplying both sides by of that equation by something else (like some number "c") is still a valid equation (a*c=b*c) even if one side is an infinitely long sum... pretty cool

[@gsparkway324]

the normal equations for a sum of a geometric series is S=a(1-r^n)/(1-r) where n = # of terms & r = multiplier this formula he uses does not involve a multiplier, so it's easier to use In the last example r=1/2 & n=infinity answers turn out the same =2

[@xPloSiVgT]

Your explanations are marvellous

[@andreasggeorgiou000]

SalMAN you 're the MAN!!!!!!!!!

[@boeing747200lr]

Sal, I find you amazing. Man, I gotto worship u.

[@lisinka3]

I think the formula S = (a^N+1)-1 / a-1 breaks down. Assume there's the same series where a = 2, k =1 and N = 3 That would equal 2^1 + 2^2 + 2^3 = 14 But with the formula it's: ((2^3+1)-1) / 2-1 = (2^4)-1/1= 16-1/1 = 15 That formula only works when k = 0.

[@aldarie]

Sweet! Tons of fun! Thanks Sal!

[@andreasggeorgiou000]

@supercalifragilismic It means that the sum of the series is a specific number (i.e. it's not equal to +∞ or -∞).

[@WorldCollections]

I trully admire you ! Anyways, I really had some trouble trying to understand (at about 3:10) how did the term a^(n-1) cancelled...it didn't even make it to the list!

[@boredbigmac]

who else is bothered by the undeleted DOT at 4:20?? anyway, thanks a lot sal!

[@Addicti0nz1]

my brain starts malfunctioning when i walk into a classroom

[@dcourtn1]

keep it up sal

[@essammqottb3857]

This has been great thank you alot

[@xToTaLBoReDoMx]

technically since there's no bracket... 1^k=1 for all values of k... So 1^k/2=1/2 for all values of k.

[@mechwarreir2]

when you were showing the infinite series, just wonderin, but what is elipson? My teacher mentioned something about a really small number about limits of sequences or something. I forgot though.

[@johnson3741]

Its fun to try and follow his mouse with mine. Wait...I'm supposed to be studying...damnit.

[@daawinkdoe9818]

This is very helpful thank you☺

[@AssyrianKing4ever]

My college teacher tells us stories and in the end of the class he gives us youtube videos to watch...Wattt a teacher ha?

[@fspopshovit]

That actually blew my mind..

[@BradleyFMash1]

1/2^infinity is so small its arbitrary and therefore in all sense of rationality, it is 0. So like he did, he just crossed it out and reduced to 2. Calculus is awesome!

[@dedly13]

draw it out? that would take a long time to draw an infinite number of terms!

[@AlphaCrucis]

Wow that's awesome. Seems strange multiplying .9R by 10 though. I'll remember this.

[@CarouselFlip]

Right; it is defined as the number of ways in which one can arrange 0 objects ;)

[@wetim99]

WTF that blows my mind man, so if we add infinite numbers to the exponent it adds up to 2?

[@isorry4strokin]

@MilitaryMan006 thats the point of the formula... it takes a number that you would otherwise be unable to obtain, and it makes it into something tangible.

[@turbold1]

can you explain me the concept of what it means if a infinite geometric series whose absolute value r

[@TheSadDuck]

You are the friggin man tho Sal

[@alder7141]

that is sooooo cool lol

[@mahletissayas6199]

That was helpful, thank you

[@marckinio]

you're wrong men, that series is a famus paradox and the infinite sum of (1/2^n) goes to 1 when n goes to infinity. Thats easy to see.

[@plataoplomo9018]

the difficulty from part one to part two is overwhelming.

[@TrainPlaneBoat]

yupp

[@Jtking3000]

how would you do the sum of the series 1+ 0.5 +0.333 +0.25 +0.2 .....1/infinity?

[@jonathanaguilar2178]

He totally ignored the negative sign in the numerator right at the end. Great Tutorial anyways!

[@eiscube6232]

+Jonathan Aguilar the negative sign cancelled out, he didn't ignore it.

[@spacehooliganzack7429]

Actually, 1/2^0 does equal 0! since 0!=1 :D

[@alberteineshtain1644]

Zachary Leib-Perry can you explain ? I thought every number to the power 0 =1

[@spacehooliganzack7429]

ahmed alamassi Yes. Exactly

[@Reydriel]

+ahmed alamassi Every number *(except 0) to the power 0 is 1. 0^0 is undefined :P

[@alberteineshtain1644]

+Reydriel yes I know.. I just didn't pay attention to the factorial notation I thought he used the elimination mark because he is exited , I didn't know it's factorial when he said "1/2^0 does equal 0! " I thought its like one half to the power zero does equal zeeeero !! so I asked him.

[@theartemys]

Yup that made me laugh too XD

[@E3tiger]

Just wondering. Why do you put your equals signs with one of the ends joining? Sort of like a c.

[@drevil37373]

last one is a limit, you would never actually reach 2, what you can do though is get like 20 other peopel go to a bar, have the first person ask for half a beer, then the next person half as much as them so on and you would of never used a full beer cause of this.

[@ninjajesus81]

So is this one of those things where it doesn't technically ever get to 2, but it's really just 1.9999999999999999999999999999999999999999999999999999999999 with infinite 9's?

[@ayesharulzz00]

r there only two parts ???

[@adam13579246]

In the annotation at 7:50 I couldn't help but think "but 0! does equal 1"

[@Waranle]

brco2003 you got some issues my friend. Sam nice video thank you

[@hen555]

how come in first example (first video) you add the sums of the reverese and forward versions ... but in this example you subtracted the sum and did a*S. How do we know what to do ?

[@purpk86]

Sal, can you please make videos involving elementary analysis? I can never find any videos :(

[@Dentei3]

Beautiful

[@bananaondrugs]

@ACfireandiceDC compared to some college/university/high school teachers/professors he's explaining it on a kindergarten level. Besides this stuff is kind of easy. I went through it a few years back and I'm 16 atm... Anyway you can always pause the video and go back. And Sal...nice job on finding your own mistakes(usually takes me hours to find that at the middle of a problem I've randomly inserted/removed bullshit/important stuff. Or occasionally trying to find out that 7=0 is wrong.

[@jasonthomas9975]

Hello..welcome back to this comment that u posted 11 yrs ago

[@jasonthomas9975]

How has ur life been

[@valdas0]

So where are the other sequences which were mentioned in the part 1?

[@josefinej3921]

But how is it that 1/2^infinite is 0? The number gets smaller and smaller, so it shouldn't reach exact 0 at all, should it? Or is it just because the last numbers get so small that we don't need to count them in at all? This reminds me of Zeno's paradox.

[@j1george]

Watch the video on limits

[@spacehooliganzack7429]

When the denominator gets extremely large the number becomes so small we might as well call it zero. But infinity is a weird place. Infinity is so monumentally colossal that it actually does make the value of 1/2^infinite=0

[@MsMarlinDash]

7:34 and it actually c- adds up :) you were thinking converges, no? ;)

[@niceperson2]

Need a video of cauchy sequence

[@xnbentari]

It makes me sad that it only goes up to 2 :(

[@yoshirochen112]

Please make some newer videos on this topic, that would help! The content is good though.

[@Jjunior130]

that is cool

[@venomxheart]

@adam13579246 Actually, anything raised to the zeroth power is going to be 1. Regardless of it's .11115^0 or infinity^0 ...

[@Mark6770]

Thanks Sal.... I've learn so many things about you.... But How can I have the equation of the sum of Consecutive Powers? in any S(k)?

[@jmacena11]

I appreciate your work, you're really good at it but the geometric series was beyond confusing. You should have edited the video properly to make it more sensible and accurate.

[@Notasthinkasudrunkim]

@MilitaryMan006 You should look up the greek philosopher Zeno's paradox about Achilles and the tortoise. :-)

[@mazhartejani]

does this infinty degree only works for 1/2 or a fraction? what if i take 2 as a number "2^k" then what will be the answer?

[@RansomUsername]

It only works for numbers less than one. The infinite sum of 2^k would diverge, since it is greater than one.

[@nineironshore]

Zeno really went twice the distance?

[@nineironshore]

Looks like we shouldn't start at k equals zero if we want the zeno sum. We should start at one otherwise the series begins with a 1.But Sal was misleading us (hopefully not intentionally) by showing zero as the first element of the sum but 1/2 to the power of 0 equals 1.

[@Alejandro-Te]

At 3:30, should it be aS-S=(a^n+1) +1? (instead of aS-S=(a^n+1) -1)

[@vu2171994]

Citriano Torres why? aS - S. what didn't get cancel out was a^n+1 and -1. so it can't be (a^n + 1) +1. think again

[@GoodVolition]

You subtract the a^0 from the a^N+1, thus the -1.

[@GoodVolition]

Assuming a != 0.

[@daniel2177]

That's the Harmonic series: E (from n=1 to infinity) 1/n... The Harmonic series diverges, therefore no Sn for you. :-)

[@lamnguyentunglam585]

and i was wondering why is n-2 what just happen , why ? who am i xD anyway good content

[@blinkblink1358]

do infinite geometric sequences only apply to numbers below 1?

[@awesomesoccerfan]

@supercalifragilismic The act of converging (coming closer).The approach of an infinite series to a finite limit.

[@j1george]

I understand the math, but I'm having trouble visualizing it. I drew the pie, but that only adds up to 1.

[@jisuahn6552]

that might be because the first term in the series should be 1, not 0.

[@j1george]

ahh. thats what i get for turning off annotations

[@SnipeSniperNEW]

last question we can know the value of the sum by using limit as you showed but how can we know the term where the answer becomes 2?

[@ZVPieGuy]

Sniper The 0 should have been a 1. (1/2)^0=1. So 1+1/2+1/4...=2

[@noblessus]

lol my brain also starts malfunctioning when I'm running out of time in a test.

[@wiiiissam]

it won't give a clear number 2, it can never reach 2, it will be very close, but it can never reach 2, in the same way that ½^n+1 can never reach 0

[@Bioquakee]

@MrSnorkl Damn! you beat me to it! Good catch xD

[@canliao1842]

is there a geometric sequence?

[@matthewjames1378]

at 4:38 why did u use a-1?

[@dennisdicicco6406]

thank you, KHANNNNNNNNNNNNNNN

[@sophieoddling]

So the series converges at 2?

[@Layla7111]

Alright. Finals here I come :u

[@zehramehdi4724]

i stan one(1) man

[@zguitarmaster]

I don't get it. On the other video you take S and sum it to another S that has the same sum inverted. On here you take the original sum and substract it to a*S. Why? What is the thought process when you derive an equation for a series?

[@USMChiLD]

I understand the math. But conceptually, I'm having trouble grasping the idea that I can sum up an infinite number of terms to end up with a finite number. How in the world does that happen? Is it cuz numbers get so small that they become irrelevant so that the terms become 0? But then again, for an infinite amount of time, all these lil tiny numbers eventually add up to create whole numbers...it is an infinite series. Reminds me of scientists saying the universe is infinite and growing...?

[@toniikaialexis]

I really thought i got it in the firs video but this video i lost all confidence.

[@annusolanki1031]

In (1/2) ^ k is written as 1^k/2

[@mich3lleeee]

I don't get why you have to do a times S and all that. Why can't you just have S like the start...

[@TheOwnerPO]

where is part 3 and 4 ?

[@microwavecoffee]

The first term in the final question is 1, not zero due to the fact that (1/2)^0 = 1 rather than 0. Wasted 5 minutes trying to figure this out haha.