Sequences and Series (part 1)

Sequences and Series (part 1)


Let’s learn a little bit
about sequences and series. So what’s a sequence? Well, a sequence is just a
bunch of numbers in some order. You know, the most difficult
sequence is 1, 2, 3, 4. You get the point. And what’s a series? Well, it’s often represented–
it’s just a sum of sequences, a sum of a sequence. So, for example, the arithmetic
sequence– sorry the arithmetic series is just the sum of
the arithmetic sequence. So 1 plus 2 plus 3 plus–
we could keep going until maybe some number. This is called the
arithmetic series. Nothing too fancy here. But before we move forward,
let’s get a notation for how we can represent these sums
without necessarily having to write out all of the digits or
having to keep doing this dot, dot, dot, plus notation. And that notation
is Sigma notation. That’s an upper case Sigma. And how do you use
Sigma notation? Well, let’s say I wanted
to represent this arithmetic series. So I would say, well,
let’s add up a bunch of– let’s call them k’s. This is an arbitrary variable. And we’ll start at k equals 1. We’ll start at k equals
1, and we’ll go to k is equal to big N. And this is the exact same
thing, so we first make k equal to 1, and then we add it to k
is equal to 2 plus 3, and we go all the way until N minus
1 and then plus N. So this is the Sigma notation
for the arithmetic series. Before I move on, I think this
is a good time just to like learn a little bit more about
the arithmetic series. We’ll actually focus on this
one and the geometric series because those are the two
that you’ll see most often. And then once you learn
calculus, I’ll show you the power and Taylor series, which
is exact– the Taylor series is a specific version
of a power series. But let’s play around with this
arith– I keep wanting to say arith-MET-ic, but a-RITH-metic,
either way– series. So let’s call the sum S. Let’s say that this is equal to
the sum from k is equal to 1 to N of k, which is equal to, just
like we said, 1 plus 2 plus 3. And we’ll just keep adding
them, dot, dot, dot, to a bunch of numbers, to big N minus
1 plus big N, right? Fair enough. Now, bear with me a second. I’m just going to write that
same exact sum again, but I’m just going to write
it in reverse order. And I think it’s intuitive to
you that it doesn’t matter what order I add up numbers in. They’ll add up to
the same number. 2 plus 1 is the same thing
as 1 plus 2, right? So let me write this exact
same sum, but I’ll write it in reverse order. So that’s the same thing as N
plus N minus 1, plus N minus 2, plus– and the pluses
keep going– plus 2 plus 1. This is the exact sum, just
in the reverse order. And I did that for a reason
because now I’m going to add both sides of this equation. I’m going to take– S plus S. Well, that’s just 2S. And that’s going to equal
this sum plus this sum. I wrote this so that
the sum becomes clean. And why do I say that? Well, let’s add up
corresponding terms. We could have added up any
terms, but– so since they all have to add up, let’s just add
the 1 plus the N, then we’ll add the 2 plus the N minus 1,
then we’ll add the 3 plus the N minus 2, and so forth. And I think you’ll see in a
second, or maybe you already realize why I’m doing this. One plus the N, the 2 plus the
N minus 1, the 3 and the N minus 2, all the way to the N
minus 1 and the 2, the N and the 1. What’s 1 plus N? Well, that’s just
N plus 1, right? What’s 2 plus N minus 1? Well, that’s also
N plus 1, right? What’s 3 plus N minus 2? I think you could guess. It’s N minus 1. And we just keep doing that. And what’s N minus 2 plus 2? Sorry, this is a plus. N plus 1. And what’s N plus 1? Well, that’s just N
plus 1, of course. So my question to you
is how many of these N plus 1’s are there? Well, there are N
of them, right? Each N plus 1 corresponds
to each of these terms, so there are N of these. So instead of just adding N
plus 1 N times, we could say that this is just
N times N plus 1. So we have 2 times the sum is
equal to N times N plus 1, and we could divide both sides by
2, and we get the sum is equal to N times N plus 1 over 2. Now, why is this neat, or
why is this cool at all? Well, first of all, we found
out a way to sum this Sigma notation up. We got kind of a
well-defined formula. And what makes this especially
cool is you can use this for low-end parlor tricks. What do I mean by that? Well, you can go up to someone
and you can say, well, how quickly do you think I can add
up the numbers between 1 and– what am I doing– oh,
between 1 and 100? And, you know, people will say,
oh, it will take you a little time: 1 plus 2 plus 3. And you say, well, it takes
me no time at all because this is what I can do. So the sum– and I just want
to show you that you can use different variables from B
equals 1, we’re taking the variable B, to 100, right? That’s the sum from 1 to 100. And we figured out
what that formula is. It’s going to be 100
times 101 over 2. Well, what’s 100 times 101? It’s just going to be 101
with two zeroes, right? 10,100 over 2, and
that equals 5,050. That’s pretty neat. Instead of having to say 1 plus
2 plus 3 plus blah, blah, blah, blah, blah, blah, blah, blah,
plus 98 plus 99 plus 100, this would take you some time, and
there’s a very good chance you would make a careless mistake. We could just plug into this
formula, which we proved and hopefully you understood,
and say that equals 5,050. You could do even something
more impressive: the sum from 1 to 1,000. What’s the sum from 1 to 1,000? Well, our formula, remember,
was N times N plus 1 over 2. So if N is equal to 1,000,
then what’s our sum? It’s 1,000 times 1,001 over 2,
which is equal to– well, we’ll just add three zeroes to
this: 1,001, one, two, three. Sorry, I think that was
my first burp ever on one of these videos. I should re-record it, but
I’m going to move forward. That kind of disconcerted
me a little bit. I’d eaten too much. Anyway, divided by 2,
and what is that? Let’s see, it’ll be 500–
let’s see, this is a million. Half of a million is 500,000. 500,500. And that would have taken
you forever to do manually. But based on this formula we
just got, you know how to do it very, very quickly. So that’s the
arithmetic series. But let’s do another one. This is another typical
series that you might see. Actually, this one you’ll see a
lot in your life, especially if you go into finance or really a
whole series of scientific– this shows up a lot, and this
is called the geometric series. And the geometric series is–
essentially you take x. And I’ll do it generally where
I just take a variable x, and I say– well, no, no. Let me just not take an x. Let me just take some number. So let’s say some number a to
the k from– I don’t know. Let’s say from k is equal
to 0 to k is equal to N. What does that mean? Well, that means a to the 0,
right, k is 0, plus a 1 plus a squared plus a to the third
plus– and you could keep going– plus a to the N minus
1 plus a to the N minus 2. This is called the
geometric series. And it might not be obvious to
you, but this type of growth, where you keep increasing the
exponent, this is called geometric growth. So how do you take
the sum of this? Well, let’s see if we can do a
similar trick, although this trick will involve
one more step. So let’s call the sum S. Let’s call it the sum from k
equals 0 to N, a to the k. And that, of course, is
equal to what I just wrote. I probably didn’t have
to do it like this. a squared plus bup, bup, bup,
bup, plus a to the N minus 1, plus a to the N minus 2. Now let’s define another sum,
and I’m going to call that aS. Actually, I’m about to run out
of time, so I’ll continue this in the next video.

100 thoughts on “Sequences and Series (part 1)

  1. I love how he says "The most typical sequence is 1,2,3,4,…" and the subtitles say "The most difficult sequence is 1,2,3,4,…"

  2. um at 9:03, it should be A raised to power N. you wrote A raised to power N-2. that should come before A raised to power N – 1. just saying.

  3. In contrast to (well, I think) the majority of people here, my teacher isn't underqualified. It's just that she overcomplicates EVERYTHING until our heads are spinning.
    This actually makes so much sense.

  4. I absolutely love your videos, and you save my life when time comes for my calc final, but I can't lie: that burp moment was my favorite part out of all of these videos hahaha.

  5. So cool! My maths teacher never told us how the formula came about! 🙂 For arithmetic series.. this is so useful…

  6. And what, they should be run by the ever-objective, omnipotent, and totally fair government? You make it seem as though the a business or a government isn't already a group of people who do in fact make money from their jobs.

    The only difference if it were privatized would be that competition would drive up quality, and drive down costs; and there would be significantly less wasted funds. Private schools spend less per student and provide far better education.

  7. I'm not for privatizing prisons, let me make that clear. However, the administration of the prison doesn't decide whether or not people are arrested. That's up to the cops — if we want to reduce our prison population, we should legalize drugs, stop targeting minorities, and most of all, stop judging a cop's performance by his/ her number of arrests, and rather by the quality of their work.

  8. legalizing drugs, namely marijuana, would actually solve a few issues. one of the main reasons weed hasn't yet been legalized is the high profit made from drug busts. the police departments and the government in general make a shit ton of money off of marijuana being illegal. and more money for the government is what the government wants. ergo, illegal weed. c:

  9. You gotta give the ending a speech bubble to correct the mistake you made (I think), the last term should be a^n, not a^(n-2). Correct me if I'm wrong ._.

  10. this is beneficial however n(n+1)/2 will only work if the numbers are incrementing by 1, if the start going up in 2's or 3's this will now work, you will have to use n/2(2a +(n-1)d) where a is the start and d is the commen difference

  11. 1n a lottery, the first ticket drawn wins a prize of $25. Each ticket after that receives a prize that is twice the value of the preceding prize.
     
    a) Write a function to model the total amount of prize money given away.
    b) How many prizes are given out if the total amount of prize money is approximately $2 million?

    Can someone help me with this question?

  12. at the end of the video, when adding the geometric series, a to n-2 comes before a to the n-1.That makes it more understandable because a to the n-2 is smaller than a to the n-1.Nice video though (y)

  13. Am I mistaken or, since k=,1 the formula should be n*(n+k)/2?
    Otherwise the formula doesn't apply to other sequences.

  14. I'm watching this before we even learn it to be prepared and… I'm scared to even start the video… O.O internally crying

  15. Why have u written (N-1),(N-2). in geometric series???
    I don't think it's right!!!!
    It must be (N-1),N.
    Isn't it??

  16. The pronunciation of "arithmetic" should be "ar-ith-met-ik" with MET stressed because the word is an adjective in this case.

  17. 20 years after my 7th grade math instructor asked us this question, i run into his solution with my Calc II work. Looking back, that guy is optimistic about 7th graders being able to figure out the Sigma property.

  18. actually the most confusing math i have ever done. anyone else understand the math but not the mathmatical terms? and why n-1? where does that crap come from???

  19. I can't understand ☹️why math is very difficult like my gf she is very difficult to understand that's why we break.

  20. someone explain what went through his mind when he decided to add the 2 series. like i get the end reason but how would he have known to add the 2?

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