Explicitly defining a seriesYouTube↗  IA↗(long shot)
Education  115.7K views 57 likes 2 dislikes   26 Nov 2013
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Thai Tee Ponsukcharoen
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Let's say that we're told that this sum right over here, where
n 이 2에서 부터 무한대로 갈 때
our index starts at 2 and we go all the way to infinity,
여기에 주어진
that this infinite series is negative 8/5
무한 급수의 합은
plus 16/7 minus 32/9 plus-- and we just
(-8 / 5) + (16 / 7) + (-32 / 9) + ...
keep going on and on forever.
그리고 끝없이 이어집니다
And so what I want to do is to explicitly define what a sub n is here.
그래서 여기서는 일반항이 무엇인지 명쾌하게 밝히는 것이 목표입니다
So right now we just say, hey, if you
그러므로
take the sum of a sub n from n equals 2 to infinity,
n 이 2에서 무한대로 갈 때 일반항의 합을 구하면
it turns out you get this sum right over here.
여기에 있는 합이 나오는 것입니다
But let's think about what a sub n-- how we can actually
하지만 지금은 일반항이 무엇인지 생각합시다
define it in terms of n.
n에 대해서 어떻게 정의할 수 있는지 말입니다
And I encourage you to pause the video right now and try it
그리고 이 비디오를 멈추고
on your own.
당장 스스로 시도해 보시길 바랍니다
So the first thing that you might realize
여기서 가장 먼저 알아차릴 수 있는 것은
is, well, this is the number that we're going to get.
이것이 얻을 수 있는 숫자인데요
Let me write it this way.
이렇게 써 보겠습니다
a sub 2 is equal to negative 8/5.
두번째 항은 (-8) / 5
a sub 3 is equal to 16/7.
세번째 항은 16 / 7
a sub 4 is equal to negative 32/9.
네번째 항은 (-32) / 9 입니다
And I'm just giving the sign to the number in the numerator.
그리고 근호는 분자에 붙어 있습니다
Negative 8/5 is the same thing as negative 8 over 5.
즉 (-8) / 5 는 5 분의 (-8) 입니다
Let me make that a little bit clearer. So I'll make that a little bit clearer.
좀 더 명확하게 쓰겠습니다
So this is negative 8/5.
그러므로 이것은 (-8) / 5 이고
Obviously, this is positive, so I
이것은 아시다시피 양수이므로
don't have to really worry about it too much.
크게 고민할 필요가 없습니다
And then here, I'm just saying negative 32/9,
그리고 이것은 (-32) / 9 로
so it's the same thing as negative 32 over 9.
9 분의 (-32) 와 같습니다
So let's see if we can first find
자 그럼 먼저
a pattern in the numerator.
분자에서 규칙을 찾아낼 수 있는 지 살펴 봅시다
So when we go from negative 8 to 16, what's happening?
(-8) 에서 16 이 되게 하려면 어떻게 해야할까요?
Well, we're multiplying by negative 2.
물론 (-2) 를 곱하면 됩니다
Now, to go from 16 to negative 32,
그럼 16 에서 (-32) 가 되게 하려면
we're multiplying by negative 2 again.
또 (-2) 를 곱하면 됩니다
So you might say, OK, well, whatever
그러면 분자에 있는 수는
we have in the numerator must be a power of negative 2.
(-2) 의 몇 제곱인 것을 알 수 있습니다
And, all right, if you say, well, maybe this
그리고 이것을
is negative 2 squared, well, you know that negative 8
(-2) 의 제곱이라고 생각했을 때
isn't negative 2 squared.
물론 (-8) 은 (-2) 의 제곱이 아닌 것을 압니다
Negative 2 squared is equal to positive 4.
(-2) 의 제곱은 4 입니다
Negative 8-- this right over here.
여기에 있는 (-8) 은
Negative 8, that is equal to negative 2 to the third power.
(-2) 의 3승과 같습니다
16 is equal to negative 2 to the fourth power.
16은 (-2) 의 4승과 같습니다
Negative 32 is equal to negative 2 to the fifth power.
(-32) 는 (-2) 의 5승과 같습니다
So notice, our exponent on the negative 2
보시다시피 (-2) 의 지수는
is always going to be one more than our index.
일반항의 n 보다 항상 1 이 큽니다
Our index is 2, our exponent is 3.
n = 2 일 때 지수는 3 이고
Our index is 3, our exponent is 4.
n = 3 일 때 지수는 4 이며
Our index is 4, our exponent is 5.
n = 4 일 때 지수는 5 입니다
So that gives a sense that at least the numerator is going
그러므로 n 이 주어졌을 때
to be-- whatever our index is, it's going to be-- so let
분자가 어떻게 나올 지 알 수 있습니다
me write this down.
그럼 적겠습니다
So a sub n is equal to-- well, it's
그러므로 일반항은
going to be negative 2 to whatever index we're at,
n 의 숫자에 따라
to that index plus 1 power.
(-2) 의 (n + 1)승일 것입니다
So that's a reasonable way to think about our numerator.
그래서 이것이 분자를 구하는 타당한 방법입니다
Now let's think about our denominators.
자 그럼 분모를 생각해 봅시다
So over-- So we go from 5, so when n is 2, we're at 5.
5부터 시작하는데 n = 2 일 때 5 이고
When n is 3, we're at 7.
n = 3 일 때 7 이며
When n is 4, we're at 9.
n = 4 일 때 9 입니다
So notice, 5 is 2 times 2 plus 1.
보시다시피 5 = (2 × 2) + 1 이고
This right over here is 2 times 3 plus 1.
이것은 (2 × 3)+ 1
This right over here is 2 times 4 plus 1.
이것은 (2 × 4) + 1 입니다
And you should just kind of play around
그리고 머리 속에서
with different patterns in your head
답이 나올 때까지
until you say, hey, well, look, this
여러가지 규칙을 생각해 보면
is increasing by 2 every time.
2 씩 늘어나고 있는 것을 볼 수 있습니다
Notice, this increases by 2 every time.
보시다시피 2 씩 증가하고 있습니다
But these aren't exactly multiples of 2.
하지만 2 의 배수라고는 말 할 수 없습니다
These seem to be off by one more than the multiples of 2, which
이 숫자들은 2 의 배수에 1 을 더한 것으로 보이는데
is a good sign that this is going
이것은
to be 2 times our index plus 1.
2 를 n 의 값에 곱한 것에 1 을 더한 것임을 보여줍니다
So we could write this as 2 times our index plus 1.
즉 2n + 1 이라고 쓸 수 있습니다
And we're done.
그리고 끝났습니다
That's what a sub n is.
이것이 일반항입니다
And if we wanted to write this series in sigma notation,
이 급수를 시그마를 사용해서 표기하고 싶다면
we would write this as the sum from n
이것을 n = 2 에서
equals 2 to infinity of negative 2 to the n plus 1 power over 2n plus 1.
무한대로 갈 때 2n + 1 분의 (-2) 의 (n +1)승 의 합으로 쓸 수 있습니다
And that would equal this series right over here.
그리고 이것이 여기의 급수와 같은 값이 될 것입니다
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