Sequences and series (part 2)

[GenericCoder]

This video is mind blowing thanks alot :D.

[dedly13]

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

[zepherighost]

I'm going to go ahead and assume that correct with the first term wasn't a zero factorial and was a marker of exclamation.

[melisantika]

I am very happy to see the vidoe Finding the sum of an infinite geometric series. from you, hopefully the others also are happy for You

[Onepissite]

I am very happy to see the vidoe after you give this Finding the sum of an infinite geometric series.

[Ondelendo]

I Love The Video Finding the sum of an infinite geometric series It Can Increase My Knowledge

[bebeheuy]

Steady I Really Like This Video Finding the sum of an infinite geometric series.

[willamricard]

Nice Video Finding the sum of an infinite geometric series That You Share , So Very Nice Thanks You

[imegatrone]

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

[anakmudajaman]

after i watched this video Finding the sum of an infinite geometric series, my insight is very open because the video is very good to give information

[M1ch43lW00d]

To infinity and beyond!

...or not.

[boredbigmac]

who else is bothered by the undeleted DOT at 4:20??

anyway, thanks a lot sal!

[MrSnorkl]

At 8:16 , that's pretty funny since 0! is still equal to 1. Definitely doesn't equal 0 though.

[Bioquakee]

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

[CynicalApple]

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

[AsianAlaskan]

Don't click the facebook icon, don't do it!

[firefox690]

Mind Fuck :P

[tedoymisojos]

Oh god XO I dont understand

[MilitaryMan006]

isn't this theoretically impossible? I mean think about it, you can't get a finite solution with an infinite amount of terms it just doesn't happen

[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.

[Notasthinkasudrunkim]

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

[MilitaryMan006]

@Notasthinkasudrunkim zeno's paradox? the one with a person chasing someone else but theoretically never reaches them? I can easily explain it. If you take an infinite amount of smaller terms, it is obviously never going to hit. But since our Universe is quantized, the smallest possible unit in space is 5.54 x 10^-35. Therefore the sum is not composed of an infinite amount of numbers. I will PM you the solution to the paradox and disprove it mathematically rather than scientifically.

[Notasthinkasudrunkim]

@MilitaryMan006 Fully aware of it's incorrectness, just though it'd shed some light on what's he's saying. Maybe not :D Got your pm and I hope the next explanation is better :-)

[azad3044]

this vedio helped a lot. first time i understood summations.

[ACfireandiceDC]

I can't interpret what he is saying. He's explaining it as if the viewer already knows how it works. I can't keep up with him.

[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.

[MeestaNoName]

@ACfireandiceDC watch part 1 then

[ACfireandiceDC]

@MeestaNoName I have the same problem with Part 1.

[tska84]

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

[Br3nD0g]

bah bah bah bahhhh

[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)?

[KeyOfAm]

Yea, it is a^n! not a^n-2 =D

[andreasggeorgiou000]

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

[supercalifragilismic]

what does convergence mean?

[andreasggeorgiou000]

@supercalifragilismic

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

[supercalifragilismic]

@andreasggeorgiou000 thanks!

[awesomesoccerfan]

@supercalifragilismic

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

[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?

[Wafffl3]

@ninjajesus81 1.999 repeating does equal 2.

[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.



[1059232]

@marckinio no, that's if the first term is 0. The first term in the example series is 1.

[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.

[G33k0fL1f3]

on the annotation"(not 0!)" the "!" should be on the outside because for a few seconds I thought it was "0 factorial".

[valdas0]

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

[RToor34]

Who else goes here because they don't pay attention in class.

[liliprincess717]

@RToor34 No. Mine just doesn't know how to teach.....

[feraligatr8]

@RToor34 haha I think I'd be better off just skipping school and watching his videos

[3333BlackWidow]

@RToor34 I`m here cause my teacher can`t teach -_-

[ynut666]

I think on the 04:10 the a^(N+1-1) should be a^(N+1) - 1 the minus 1 should not be in the exponent. It is just an integer. But the video is very helpful... thanks.

[K13ization]

i have a question that a ball dropped from 3 metres rebounds to a height 1.5 times of the previous height and what would be the total distance it would cover? so 1.5 square is obviously greater than 1.5 so how would i find the distance of infinite bounces ?the answer given in the book is

except first 3 remaining series is infinite G.P

sum of infinite G.P = a/(1-r)

sum will be 3+3/(1-1/2) = 3 + 6 = 9 m

cant understand a thing please help

[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 ?

[hen555]

i think in reality it is not a finite number of 2... but 2 decimal like 2.0000000000000000000000000000­000000000000000000000000000000­000000000000000001 or somthin? i dunno since the number is approacaching 0 in the limit, it is not realy 0 just really close... infinitlly close?

i dunno this is just what i think lol is it true?

[cytocarbon]

Where's part three? I need to know ∑(n=1)^∞(cos(nπ)/(n√n))

[johnson3741]

Its fun to try and follow his mouse with mine.

Wait...I'm supposed to be studying...damnit.

[DOUBLECHEESEBARGER]

Fire all the teachers.Electronically transfer every penny these teachers all around the world going to earn into Kkhan's account, and piss all the teacher off to their very core.Actually, i don't get it. I have never seen a teacher good enough in one subject; however, i see one dude, probably well underpaid dude,doing everything taught in school. How Khan? How? thnx very much.But i can't effectively navigate through the videos though. does he do every section orwhat?1eacher,7continents,7bi­l stud

[adam13579246]

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

[TheDGFproductions]

@adam13579246 wow thats cool, thanks for pointing that out 0! = 1

[venomxheart]

@adam13579246

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

[adam13579246]

@venomxheart You are correct, now could you please point out where I mentioned anything about exponents in my previous comment -.-

[whattheheck102]

@adam13579246 Bahahahahah! A+ for the math humor.

[Hudstahh93xoxo]

@adam13579246 He is not talking aout factorial but just happened to use an exclamation mark

[Patch97236]

patrickJMT delves more extensively into Sequences and Series, check him out as well. Both Khan and Patrick are really great with their videos. Thanks!

[Patch97236]

patrickJMT delves more extensively into Sequences and Series, check him out as well. Both Khan and Patrick are really great with their videos. Thanks!

[jshields34]

lol at 9:00 he draws a penis

[TheDGFproductions]

@jshields34 save that humor for our boring ass classrooms

[Muffinfordinner]

In all arithmetic examples he uses k = 1 and

for geometric examples he uses k = 0

Does this have to be the case? And how would you use the formulas if it weren't?

[krajtl]

erroer

[dcourtn1]

keep it up sal

[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

[EsHammer1001]

Oops, I just realized my mistake. I forgot to include a^0, which makes both my previous comments wrong.

[EsHammer1001]

Furthermore, (1 / a^k) appears to converge on (1 / a - 1). So if a = 2, then the result would be (1 / 2 - 1) or (1 / 1) and it converges on 1, not 2. If a = 3 then the result converges on (1 / 3 - 1) which ends up being 1/2. Likewise (1 / 4^k) converges on 1/3. These are all through observation using actual numbers however, and I do not have the proofs for them at this time.

[EsHammer1001]

You wrote that the formula around 5:50 comes out to be (a^(N+!) - 1)/a - 1, but when I actually calculate things out they seem to come to ( (a^(N+!) - 1) / a - 1 ) -1. I did this because the earlier portion did not make sense to me. Am I wrong here? Or could you show how to arrive at this mathematically, or at least show how I am mistaken? I want to tell you that I enjoy your videos immensely, thank you.

[boeing747200lr]

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

[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!

[ScorchinBeats]

wait up 1/2^n is NOT equal to zero it's equal to 0.1*10^-infinity lol

[ace1dominant1]

later on in the video at like 8:22, you said that that "this should be a 1 since 1/2^0=1 and not 0! but zero factorial does still equal to one lol

[N3rd0fa11ag3s]

is there a geometric sequence?

[MajDigi]

@MajDigi Sorry, just a glitch.

[libbern]

My view on reality got wripped apart when I learned that the sum of the arbitraly large sequence 1/n doesnt konverge^^ Seriously, that gotta be wrong:O

[InstrumentalEcstasy]

to infinite and beyond

[InstrumentalEcstasy]

would you like to do my exam tomorow

[hmdlamin93]

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

[joshuaspolar]

wow i was looking for the Limit portion and this very well explained it for me!

thanks so much!

[Addicti0nz1]

my brain starts malfunctioning when i walk into a classroom

[E3tiger]

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

[MelanholicCarnage]

It seems like geometric series are useful to graph to see physically how the infinite series adds to a real number. Cool stuff.

[BooganaMonkey]

yeah i still hate sequences =.= !!!! but not bad ... you make it quite interesting xD

[Tarkvinius]

@those with conceptual issues:

(1/3) = 0.3333... (3 repeating), agreed?

now multiply both sides by 3

the futher you chase the repeating 9 that results, the closer you get to 1.

mathematically, 0.999... is equal to 1, because the difference between this 0.999... number and 1 becomes infinitely small the more closely you 'examine' it. if you examined it to infinity...

0.999... doesn't SEEM like it should be 1 at first glance, but this is just an artifact of the decimal system

[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.

[vashfish]

sorry... meant sum of 1/2 ^ k

[soniaangel00]

thank you so much for posting these videos, its March and my A levels are in May and i'm no where near prepared and i was thinking 'omg, i'm so gonna fail' but thanks to your videos, i might do something after all lol..but truly, thank u

[AtheismIsWiked]

Wow we had to know another whole formula for infinante geometric series =[

[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!!!!

[DanielKovach]

WOOOHOOOOOOOOOO

[noblessus]

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

[rikkeoanna]

I know its stupid, but 0! factorial is defined as 1.

[CarouselFlip]

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

[maksman93]

you're right however i dont think he meant "0!" as a factorial, he just meant it as an exclamation of "not 0"

[coolbluegatorade6225]

its not actually 2 ... its veryyyyyyyyyyy close to 2 becuz the limit isnt actually 0 its verrrrrrrry close to 0.

[ichinarukurumaki]

do u know what a limit is dude?

[prajalpa]

well. you get 1.999... (repeating) which is equal to 2 :-p

[BajanCreation]

hey man im not stupid but this seems really hard to understand. Im doing banking and finance and theres a module that has this stuff. I gotta spend alot of time with this. damn i thought i knew maths.

[sweetm5789]

If r = 4x over 3 + x^2, find all the values of x for which the geometric series will converge.

I really need help with this before tuesday

[wholenote654456]

if you used an integer instead of a proper fraction, it would be a very different, and very large sum.

[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...?

[seandavidr]

It doesn't ever actually reach a finite number you just get infinitely close that you can round it to a finite number. imagin going from 1 to 0 by multiplying by 1/2 an infinite amount of times you'll get infintly close to 0 but you can never actually reach 0.

[DanielKovach]

@USMChiLD thats why your in the Marines!

[DoorKnobOwns]

@DanielKovach - That's pretty damn ignorant to say. Some of the most intelligent and level headed people I know are Marines. Have some respect for our warriors.

[alkalait]

consider you are adding numbers for ever, but each number is half of the previous one. you are never going to reach +1 of your first number... but instead you are "converging" towards it. simple huh?

[dirtybitbucket]

1. Some comments refer to the formula for a general geometric series. The one in the video is a special case, in which the first term is 1. lisinka3 tries to apply the special case formula to the general case, so of course it breaks down.

2. daniel2177 is correct, attackoftheemos is not. The harmonic series does diverge to infinity. It is true that the terms have 0 as limit, but the sum obviously cannot be 0 since all the terms are positive. Let's not confuse series with sequences.

[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.

[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

[Jtking3000]

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

[daniel2177]

That's the Harmonic series:

E (from n=1 to infinity) 1/n...

The Harmonic series diverges, therefore no Sn for you. :-)

[attackoftheemos]

The sum would go to zero, as the denominator goes to infinity, making the fraction infinitely small.

[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.

[moreoriginalusername]

The actual formula, for any geometric sequence summation is a(r^(N+1) - r^(k))/ (r-1) where a = some constant, r = the base number, and N and k are used as above. his formula works when k = 0 because anything to the zeroeth power is 1.

[SlimShuvro]

All Praises due to Salman Khan. Sal you are the effing man! I don't know if you read these comments or not but people worship you. I started a calculus 2 class very late and was completely lost in class because I had missed several topics. But you my friend, you took care of all of that. If I were a girl I would make love to you day and night.

[Pandaddy47]

You officially made math fun.

Most teachers just teach, HEY this is the geometric series, memorize. and it usually comes into one ear and out the other. Now that you taught me this, i dont think i can forget it. Major props my friend, major props!

[huge1234]

wern't you right in the first place? I thought in geometric series the powers were equal to (n-1), so the last power can't be n?

[Poke4Poker]

its not really two its more like 1.9999999999999999999999999999­999999999999999999999999 =p

[pilsburryfroboy]

Poke4poker, we actually talked about this in my math class. you can prove that .9 repeating actually equals 1 (1.9 repeating = 2, 2.9 repeating = 3.......). If you say that N=.9R (repeating)

multiply both sides by 10 (10N=9.9R)

subtract N (10N-N=9.9R - .9R) (we said N=.9R)

divide both sides by 9 (9N / 9 = 9 / 9)

and then you have N=1, but wait.... we said N= .9R :D :D kool stuff

[AlphaCrucis]

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

[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

[bluewhalesarenice]

i love sal. i'm so glad he exists and is kind and does this!! great job!

[NobleGatekeeper]

OWNED!!!

[Mizumi17]

man..... can you take my first test for me?? Its this fridaaaay plz!!! lol. this is all just so confusing at first. i kno it'll get better, but so far its just overwhelming.

[bourbondream]

3^11-1 divided by 2 = 88573

[carsanco44]

Thanks, Sal. As usual, I love how you just keep on making things seem so simple. Don't ever change! I'm off to find out if I can fully understand Taylor and Maclaurin which were always just a bit nebulous to me. I'll check and see if you've explained them anywhere. Carsanco:)

[aldarie]

Sweet! Tons of fun! Thanks Sal!

[Dippazncream]

now i gettit. L = a+(n-1)d not S.

[Dippazncream]

i was taught that S= a+(n-1)d and i have an exam tomorrow. wt da hell man im screwed.

[Valefarous]

Sylvanus Thompson in Calculus Made Easy showed a similar proof which I thought was kind of neat.

Say 1+1/2+1/4+...1/infinity = x,

multiply both sides by 2, you get

2x = 2+1+1/2+1/4+....1/infinity

Notice 1+1/2+1/4 +...1/infinity, the original series, is after the two. Now, x = the original series, so take x from both sides (as x on one side and as the series on the other):

x = 2 + series - series

Therefore x = 2.

[kypronite]

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

[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?

[thecrowlord]

Allowing 1/2^n+1 to hit zero seems right for practical or causal applications but maybe not for all logical-thought experimental ones concerning "ever and ever smaller". In which case, my brain loops into thoughts of unknowns/@&@*^@^@#$%@#&*%^*#%*­:(

[thecrowlord]

hmm.

[Waranle]

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

[brco2003]

too many mistakes

[brco2003]

I didn't mean with Sal; I meant with the questions. Great vid., Sal!

[turbold1]

can you explain me the concept of what it means if a infinite geometric series whose absolute value r<1 converges to the sum of S and what it means for the divergent. i dont get it :{ ty for the video and showing how u derived formulas. besides im gonna start algebra after this summer guess im getting head start for algebra2

[Fortinbras85]

Thanks for the videos! Its a great refresher now that I'm in higher math courses. I was wondering what the intuition is for multiplying the geometric series by a to get the formula.

[runningmarvel]

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

keep it up, i like it!

[turbold1]

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

[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!

[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.