
Bulgarian: 
Даден е безкраен ред, което е същото като сумата
на членовете на безкрайна редица S,
която е равна на сбора 
от n = 1...
искам да го запиша 
по-прегледно.
от n = 1 до n  равно на безкрайност,
от а с индекс n.
Това е малък преговор.
Можем да кажем, че това
е равно на
а с индекс 1 плюс а с индекс 2,
плюс а с индекс 3,
и можем да продължим
да събираме членовете 
до безкрайност.
Сега искам да те 
запозная с понятието
частична (парциална) сума.
Това тук е безкраен ред или сбор на 
членовете на безкрайна редица.
Но можем да дефинираме
частична сума,
ако имаме S с индекс 6,
този начин на записване
означава, че ако S е безкраен ред,
S с индекс 6 е частична сума
от първите шест члена.
В този случай това ще бъде...
няма да продължим
да събираме до безкрайност,
а това ще бъде а1 +

English: 
- [Voiceover] Let's say that you have
an infinite series, S, which
is equal to the sum from n equals one,
let me write that a little bit neater.
n equals one to infinity
of a sub n.
This is all a little bit of review.
We would say, well this is the same thing
as a sub one, plus a sub two,
plus a sub three, and we would just
keep going on and on and on forever.
Now what I want to introduce to you
is the idea of a partial sum.
This right over here
is an infinite series.
But we can define a partial sum,
so if we say S sub six,
this notation says, okay,
if S is an infinite series,
S sub six is the partial
sum of the first six terms.
So in this case, this is going to be
we're not going to just
keep going on forever,
this is going to be a sub one,

Korean: 
여러분이
무한급수를 가지고 있다고 합시다
n이 1이면서,
이 것을 조금 더 깨끗하게 쓰겠습니다
n이 1에서 무한대까지 나아갈 때의
an이라고 합시다
이 것은 약간의 복습입니다
우리는 이 것이 a1과 a2를
그리고 a3를 더한 것과
같다고 말한 것이며 우리는 이 규칙을
영원히 반복해 나아갈 것입니다
그래서 오늘 제가 여러분께 소개하려는 것은
바로 부분합이라는 것입니다
여기에 써있는 것은 무한급수입니다
그러나 우리는 부분합을 정의할 수 있습니다
그래서 S6이라고 할 때
이 표시는 의미합니다
만약 S가 무한급수라고 가정했을 때
S6는 1항에서 6항까지의 부분합을 의미합니다
그래서 이 경우 이 부분합은
우리는 이 것을 영원히 하고 있지 않을 것입니다
이 것은 a1

Thai: 
สมมุติว่าคุณมี
อนุกรมอนันต์ S ซึ่ง
เท่ากับผลบวกจาก n เท่ากับ 1
ขอผมเขียนให้สวยหน่อยนะ
n เท่ากับ 1 ถึงอนันต์
ของ a ห้อย n
นี่เป็นการทบทวนเล็กน้อย
เราบอกว่า อันนี้เหมือนกับ
a ห้อย 1, บวก a ห้อย 2,
บวก a ห้อย 3, แล้วเราก็
ทำต่อไปเรื่อยๆ
ทีนี้ สิ่งที่ผมอยากให้คุณรู้จัก
คือแนวคิดเรื่องผลบวกย่อย
ตัวนี้คืออนุกรมอนันต์
แต่เรานิยามผลบวกย่อยได้
ถ้าเราบอกว่า S ห้อย 6
สัญลักษณ์นี้บอกว่า โอเค
ถ้า S เป็นอนุกรมอนันต์
S ห้อย 6 ก็คือผลบวกย่อยของ 6 เทอมแรก
ในกรณีนี้ ตัวนี้จะ
เราจะไม่บวกตลอดไป
อันนี้จะเป็น a ห้อย 1

Thai: 
บวก a ห้อย 2 บวก a ห้อย 3
บวก a ห้อย 4 บวก a ห้อย 5
บวก a ห้อย 6
ผมทำให้มันจับต้องมากกว่าก็ได้ถ้าต้องการ
สมมุติว่า S อนุกรมอนันต์ S
เท่ากับผลบวกจาก n เท่ากับ 1
ถึงอนันต์ของ 1 ส่วน n กำลังสอง
ในกรณีนี้ มันจะเท่ากับ 1 ส่วน 1 กำลังสอง
บวก 1 ส่วน 2 กำลังสอง
บวก 1 ส่วน 3 กำลังสอง
และเราก็ทำต่อไปเรื่อยๆ ได้ตลอดไป
แต่ S ห้อย --
ผมควรใช้สีเดียวกัน
S --
ผมบอกว่าจะไม่เปลี่ยนสี แต่ผมไม่ได้เปลี่ยน
S ห้อย 3 จะเท่ากับอะไร?
ผลบวกย่อยของสามเทอมแรก
ผมแนะนำให้คุณหยุดวิดีโอนี้
แล้วลองทำด้วยตนเอง
มันจะเท่ากับ

Bulgarian: 
а2 + а3 + а4 + а5 + а6.
Мога да го направя
малко по-разбираемо.
Да кажем, че този безкраен
ред S е равен
на сбора от n = 1
до безкрайност от 1/n^2.
Тук ще бъде 1 върху
1 на квадрат,
плюс 1 върху 2 на квадрат,
плюс 1 върху 3 на квадрат,
и можем да продължим
така до безкрайност.
Но тук в тази сума...
Ще използвам същия цвят.
Тук S...
Казах, че ще сменя цвета,
а не го направих.
На колко ще е равно
S с индекс 3?
На частичната сума на
първите три члена.
Препоръчвам ти
 да спреш видеото
и да го решиш самостоятелно.
Това ще бъде просто

English: 
plus a sub two, plus a sub three,
plus a sub four, plus a sub five,
plus a sub six.
And I can make this a little
bit more tangible if you like.
So let's say that S,
the infinite series S,
is equal to the sum from n equals one,
to infinity of one over n squared.
In this case it would
be one over one squared,
plus one over two squared,
plus one over three squared,
and we would just keep going
on and on and on forever.
But what would S sub --
I should do that in that same color.
What would S --
I said I would change color, and I didn't.
What would S sub three be equal to?
The partial sum of the first three terms,
and I encourage you to pause the video
and try to work through it on your own.
Well, it's just going to be

Korean: 
더하기 a2, 더하기 a3
더하기 a4, 더하기 a5
더하기 a6가 될 것입니다
여러분이 원하신다면 조금 더 실체적으로 말씀드리겠습니다
무한급수 s가
n이 1에서 무한대로 나아갈 때
1/n²이라고 가정해봅시다
이 경우에는 1/1²
더하기 1/2²
더하기 1/3²
그리고 우리는 이 것을 영원히 할 것입니다
그러나 S는
이 것을 같은 색으로 나타내지 않겠습니다
S가
제가 색깔을 바꾸지 않았군요
그렇다면 S3이 무엇이 될까요?
1항부터 3항까지의 부분합은,
저는 여러분이 잠시 이 비디오를 멈추고
여러분 스스로 해결해보았으면 좋겠습니다
그래서 이 것은

Thai: 
เทอมแรก 1, บวกเทอมที่สอง
1/4 บวกเทอมที่สาม 1/9
จะเท่ากับผลบวกของสามเทอมแรก
และเราหาได้
นั่นคือดูว่าคุณมีตัวส่วนร่วมตรงนี้ไหม
มันจะเป็น 36
มันจะเป็น 36/36
บวก 9/36 บวก 4/36
อันนี้จะเท่ากับ 49/36
49/36
ประเด็นของวิดีโอนี้
คือให้คุณซาบซึ้งแนวคิดเรื่องผลบวกย่อย
และสิ่งที่เราจะเห็นคือว่า
คุณเขียนผลบวกย่อย
โดยใช้พีชคณิตได้
ตัวอย่างเช่น
ตัวอย่างเช่น ลองหา
ที่เขียนเพิ่มหน่อยตรงนี้
สมมุติว่า ลองกลับไปที่
เรามีอนุกรมอนันต์ S
เท่ากับผลบวกจาก n เท่ากับ 1
ถึงอนันต์ของ a ห้อย n

Korean: 
1항인 1 더하기 2항인 1/4
더하기 3항인 1/9이 됩니다
그리고 이는 1항에서 3항의 합이 됩니다
그리고 이 곳에 공통분모를 가지느냐의 여부에 따라
우리는 이 값을 알 수 있습니다
이 것은 36이 될 것입니다
36/36 더하기 9/36
더하기 4/36
그래서 이는 49/36이 될 것입니다
49/36
이 전체의 비디오에서
이 부분합의 개념에 감사합니다
그리고 우리는 부분합들이
대수적인 형태로 나타내지는
것을 보게 될 것입니다
예를 들어
우리가 스스로 실질적인
값을 여기에 나타내도록 합시다
우리가 다시 뒤로 돌아가서
무한급수인 S,
n이 1에서 무한대로 나아갈 때
an이라고 해봅시다

Bulgarian: 
първия член плюс
втория член,
1/4, плюс третия член, 1/9,
това е сборът от първите
три члена,
което можем да пресметнем.
Да видим колко е общият
знаменател,
той е 36.
Това е 36/36,
плюс 9/36 плюс 4/36.
Това е 49/36.
Целта на това видео е
да разберем какво е частична сума.
И това, което виждаме,
е, че можем да изразим
какво е частична сума
с алгебрични средства.
Например... само
да си направя малко място.
Нека да имаме бекраен ред S,
който е равен на сбора от n = 1
до безкрайност на а с индекс n.

English: 
the first term one, plus the second term,
1/4, plus the third term, 1/9,
is going to be the sum
of the first three terms,
and we can figure that out,
that's to see if you have
a common denominator here,
it's going to be 36.
It's going to be 36/36,
plus 9/36, plus 4/36,
so this is going to be 49/36.
49/36.
So the whole point of this video,
is just to appreciate this
idea of a partial sum.
And what we'll see is,
that you can actually
express what a partial sum
might be algebraically.
So for example,
for example, let's give
ourselves a little bit
more real estate here.
Let's say, let's go back to just saying
we have an infinite series, S,
that is equal to the sum from n equals one
to infinity of a sub n.

Thai: 
และสมมุติว่าเรารู้ว่าผลบวกย่อย
S ห้อย n ผลบวกของ n เทอมแรก
ของอนุกรมนี้ เท่ากับ n กำลังสอง
ลบ 3 ส่วน
n กำลังสามบวก 4
เพื่อเป็นการทบทวน
อันนี้บอกว่า
S ห้อย n --
S ห้อย n เท่ากับ a ห้อย 1
บวก a ห้อย 2 บวกไปเรื่อยๆ
จนถึง a ห้อย n
และมันจะเท่ากับเทอมนี้
n กำลังสองลบ 3
ส่วน n กำลังสามบวก 4
ทีนี้ จากค่านั้น ถ้ามีคน
เดินมาเจอคุณแล้วถามว่า
โอเค เธอรู้สัญลักษณ์
ผลบวกย่อยแล้ว ฉันมีคำถาม
ให้เธอ
ถ้า S เป็นอนุกรมอนันต์
และผมเขียนในรูปทั่วไปมากๆ ตรงนี้
S ก็คืออนุกรมอนันต์
จาก n เท่ากับ 1 ถึงอนันต์ของ a ห้อย n

Bulgarian: 
Да кажем, че знаем, че
частичната сума,
Sn, която е сборът на
първите n члена,
е равна на (n^2 – 3)
върху n^3 + 4.
Само да си припомним
какво означава това.
Sn е същото нещо като
а1 + а2 +...
и така продължаваме
чак до аn,
и това е равно на този израз,
(n^2 – 3)/(n^3 + 4).
Като знаеш това, ако някой
те срещне на улицата и попита:
"Щом знаеш как се записва
частична сума, ето един въпрос:
Ако S е безкраен ред,
записвам го в общия случай,
S е безкраен ред
от n = 1 до безкрайност
на a с индекс n

Korean: 
그리고 우리가 부분합에 대해서 안다고 가정해봅시다
Sn이 n^3 더하기 4
분의
n^2 빼기 3
이라고 가정해봅시다
약간의 기억을 되새겨보자면
이 것은
Sn
Sn은 a1 더하기
a2 더하기 , 이를 계속 해서
an이 될 때까지 하는 것입니다
그리고 이 일을 하는 것과 같게 될 것입니다
n^3 더하기 4 분의
n^2 빼기 3
그래서 만약 길에서 어떤 사람이 와서
여러분한테 말하기를
여러분이 이제 부분합의 표기법을 알았으니
여러분에게 물어볼 질문이 있다고
해봅시다
만약 S가 무한급수인 경우에는
여기에는 매우 일반적인 형태로 적어놓았습니다
그래서 무한급수인 Sn은
n이 1에서 무한대로 나아갈 때 an과

English: 
And let's say we know the partial sum,
S sub n, so the sum of the first n terms
of this is equal to n squared
minus three over
n to the third plus four.
So just as a bit of a reminder
of what this is saying.
S sub n...
S sub n is the same thing as a sub one,
plus a sub two, plus you keep going
all the way to a sub n,
and that's going to be
equal to this business,
n squared minus three
over n to the third plus four.
Now, given that, if someone were
to walk up to you on the street and say,
okay now that you know the notation
for a partial sum, I
have a little question
to ask of you.
If S is the infinite series,
and I'm writing it in very
general terms right over here,
so S is the infinite series
from n equals one to infinity of a sub n,

English: 
and the partial sum, S sub n,
is defined this way,
so someone, they tell
you these two things,
and then they say find what
the sum from n equals one to six
of a sub n is, and I encourage you
to pause the video and
try to figure it out.
Well, this is just going to be a sub one,
plus a sub two, plus a sub three,
plus a sub four, and when I say sub
that just means subscript,
plus a sub five, plus a sub six,
well that's just the same thing
as the partial sum, this
is just the same thing
as the partial sum of the first six terms
for our infinite series.
It's just going to be the
partial sum S sub six.
And we know how to algebraically evaluate
what S sub six is.
We can apply this formula
that we were given.
S sub six is equal to, well,
everywhere we see an n,

Thai: 
และผลบวกย่อย S ห้อย n
กำหนดมาอย่างนี้
เขาถามคุณว่า
เขาบอกว่า จงหาว่า
ผลบวกจาก n เท่ากับ 1 ถึง 6
ของ a ห้อย n เป็นเท่าใด ผมแนะนำ
ให้คุณหยุดวิดีโอนี้แล้วหาดู
อันนี้จะเท่ากับ a ห้อย 1
บวก a ห้อย 2 บวก a ห้อย 3
บวก a ห้อย 4 แล้วเวลาผมบอกว่าห้อย
มันหมายถึงตัวห้อย
บวก a ห้อย 5 บวก a ห้อย 6
นั่นก็เหมือนกับ
ผลบวกย่อย อันนี้ก็เหมือนกับ
ผลบวกย่อยของ 6 เทอมแรก
ของอนุกรมอนันต์
มันจะเท่ากับผลบวกย่อย S ห้อย 6
เรารู้วิธีหาค่า
S ห้อย 6 ด้วยพีชคณิต
เราใช้สูตรที่เราได้มาได้
S ห้อย 6 เท่ากับ
ทุกที่ที่เราเห็น n

Bulgarian: 
а частичната сума Sn,
дефинирана по този начин,
и ако някой ти даде
тези две неща,
и после поиска да намериш
колко е сумата от n = 1 до n = 6
от а с индекс n –
като те насърчавам
да спреш видеото 
и да опиташ да го решиш.
Това тук ще бъде а1,
плюс а2 плюс а3,
плюс а4 плюс...
когато казвам индекс,
това означава това число,
записано тук отдолу,
плюс а5 плюс а6,
което е равно на 
частичната сума
на първите шест члена на
безкрайния ред.
Това е частичната сума
S с индекс 6.
Ние знаем как да решим
алгебрично колко е S6.
Можем да използваме
дадената ни формула,
S6 е равно на...

Korean: 
그리고 부분합인 Sn은
여기에 바로 정의가 되어있습니다
그래서 여러분에게 이 두가지를알려준 사람은
여러분에게 n이 1에서 6으로갈 때의
an의 합을 구하라고 합니다
그리고 저는 여러분께
잠시동안 비디오를 멈추고 직접 찾아보시길 권유합니다
그래서 이 것은 a1 더하기
a2 더하기 a3 더하기
a4 그리고 제가 an이라고 쓸 때의
n은 그저 아래에 쓴 숫자입니다
더하기 a5 더하기 a6
그리고 이 것이 부분합과 같은 것이고
이 것은 무한급수에서의
1항에서 6항까지의 합을
나타내는 말입니다
이 것은 부분합 S6와 같게 됩니다
그리고 우리는 대수적으로 S6의 값을
구할 수 있습니다
우리는 우리가 받은 공식을
S6는 ,
우리가 an n을 어디에서 보던간에

Bulgarian: 
всяко n в нея заместваме с 6,
и получаваме 6^2 – 3
върху 6^3 + 4.
Колко е това?
6 на квадрат е 36, минус 3,
това е 33.
6 на трета степен е
36 по 6, винаги го забравям,
в ума ми изскача 216,
но нека да проверим дали
е толкова.
6 по 30 е 180,
плюс 36, да, това е 216.
Неволно съм го запомнил,
като съм срещал 6 на трета
много пъти в живота ми,
съм го запомнил несъзнателно.
6 на трета, изобщо не
пречи да го знае човек.
Значи това е 216 плюс 4,
което е 220.
Така S6, или сборът на
първите шест члена
на този безкраен ред тук,
е 33/220 и сме готови.
Смисълът на всичко това
беше просто един вид

English: 
we replaced with a six,
it's going to be six squared minus three
over six to the third
plus four, so what is this going to be?
Six squared is 36 minus three,
so that's 33,
and six to the third, let's see,
36 times six, I always forget,
my brain wants to say 216,
but let me make sure that
that's actually the case.
Six times 30 is 180,
plus 36, yes, it is 216,
so I guess I have, inadvertently,
by seeing six to the third so many times
in my life, I have inadvertently memorized
six to the third power,
never a horrible thing
to have that in your brain.
So this is going to be 216 plus four,
so 220.
So, S sub six, or the sum
of the first six terms
of the series right over here,
is 33/220, and we're done.
And the whole point of this is just so you

Korean: 
n을 6으로 교체합니다
이 것은 6^2 -3/6^3+4
와 같게 됩니다
그리고 이 것은 무엇과 같게 될까요?
6^2는 36이고 -3을 하게 되면
이 것은 33이 됩니다
그리고 6^3은
36에 6을 곱한 것이 되는데
저는 언제나 저의 뇌가 216이라고 말하는 것을 까먹네요
그러나 이 것이 확실한지 확인해보겠습니다
6곱하기30은 180이 되고
여기에 36을 더하면 216이 됩니다
그리고 제가 무심코 추측하기로는
제 삶 안에서 6^3을 너무나 많이 보아서
저는 이 것을 무심코 6^3을 기억하게 되었고
이는 여러분의 뇌에
나쁜 영향을 미치지는 않습니다
그래서 216 더하기 4가 나와서
220이 됩니다
그래서 S6, 혹은 1항부터 6항까지
여기있는 급수의 합은
33/220이 됩니다
여기에 있는 전체 내용은 이 것이 끝입니다

Thai: 
เราก็แทนที่ด้วย 6
มันจะเท่ากับ 6 กำลังสองลบ 3
ส่วน 6 กำลังสาม
บวก 4 อันนี้จะเท่ากับอะไร?
6 กำลังสองได้ 36 ลบ 3
เป็น 33
แล้ว 6 กำลังสาม ลองดู
36 คูณ 6 ผมลืมตลอด
สมองผมอยากบอกว่า 216
แต่ขอดูให้แน่ใจว่ามันเป็นจริง
6 คูณ 30 ได้ 180
บวก 36 ใช่ มันคือ 216
ผมว่า
ผมเห็น 6 กำลัง 3 หลายครั้งแล้ว
ผมจำได้เอง
ว่า 6 กำลัง 3 คืออะไร ไม่ใช่
เรื่องน่ากลัวที่มันติดอยู่ในหัวคุณ
อันนี้จะเท่ากับ 216 บวก 4
ได้ 220
S ห้อย 6 หรือผลบวกของ 6 เทอมแรก
ของอนุกรมตรงนี้
คือ 33/220 เราก็เสร็จแล้ว
และประเด็นของวิดีโอนี้คือให้คุณ

Korean: 
이 부분합과 이 것이 무엇인지
알게 된 것에 대해 감사와 비슷한
아니 혹은 진짜로 감사합니다

Bulgarian: 
да се запознаеш с този начин
на записване на частична сума
и да разбереш какво
всъщност означава.

Thai: 
ซาบซึ้ง เข้าใจ
สัญลักษณ์ผลบวกย่อยนี้
เข้าใจความหมายของมันจริงๆ

English: 
kind of appreciate, or
really do appreciate
this partial sum notation,
and understand what it really means.
