
Bulgarian: 
Както казах и в предходното 
видео,
вече сме виждали много
примери, в които започваме
с развиването на
геометричен ред,
а после приемаме, че 
частното му,
че абсолютната стойност на
на частното е по-малка от 1,
и намираме съответната 
сума.
Доказахме тази формула
в предишните клипове.
Но сега ще подходим
по обратния път.
Нека да имаме някаква 
функция...
да кажем, че h (х) е равно
на 1/(3 + х^2)...
и да се опитаме да
я представим в този вид.
След като я представим 
в този вид на сума на геометричен ред,
можем да помислим какво
е частното.
И после да я представим
като геометричен ред.
Насърчавам те да
спреш видеото на пауза
и да опиташ да го направиш.
Да видим, първото нещо,
което вероятно ти прави впечатление,
че тук имаме 1 вместо 3.
Да се опитаме да изнесем
пред скоби 3.
Това е равно на 1върху 
3 по (1 + (х^2)/3).
И понеже не искаме
тази тройка в знаменателя,

Polish: 
W poprzedniej prezentacji
widzieliśmy wiele przykładów startując
od rozwiniętych postaci szeregów geometrycznych
następnie przyjmując że ich iloraz
że bezwzględna wartość tego ilorazu jest mniejsza niż 1,
dowiadując się jaka może być w związku z tym suma takiego szeregu.

Korean: 
 
우리가 저번 영상에서 이야기 했던대로
우리는 많은 예시를 봐 왔습니다
기하급수에서
공비를 구하고
그 공비의 절대값이 1보다 작을 때
그 총합이 무엇이 될지에 대해 말이죠
우리는 저번 영상들을 통해 이 공식을 증명했습니다
이번에는 다른 방향으로 접근해 봅시다
함수를 잡아봅시다
h(x)=1/(3+x^2)라 하고
이제 이 식을 여기에 대입해 봅시다
대입하면
우리는 a와 공비의 값을 생각해 볼 수 있습니다
이제 이를 바탕으로 실제 기하급수로 나타내 봅시다
동영상을 잠시 멈추고
지금 직접 해보는 것을 추천드립니다
자, 이제 봅시다
아마 여러분이 가장 먼저 눈치채셨을 것은
a가 1이 아니라 3이라는 것입니다
그럼 3으로 묶어 봅시다
그래서 이 식은 1/3(1+x^2/3)가 됩니다
그리고 분모에 3이 있으면 안되니까

Thai: 
 
เราได้พูดถึงไปในวิดีโอที่แล้ว
เราได้เห็นตัวอย่างเริ่มต้น
ด้วยอนุกรมเรขาคณิตกระจายออกมา
แล้วสมมุติว่าอัตราส่วนร่วมของมัน
ว่าค่าสัมบูรณ์ของอัตราส่วนร่วมน้อยกว่า 1
เราหาว่าผลบวกเป็นเท่าใด
เราได้พิสูจน์ด้วยสูตรนี้ในวิดีโอก่อนๆ
แต่ตอนนี้ ลองทำกลับกันบ้าง
ลองนำฟังก์ชันมา -- สมมุติว่า
h ของ x เท่ากับ 1 ส่วน 3 บวก x กำลังสอง --
ลองเขียนมันให้อยู่ในรูปนี้กัน
แล้วเมื่อเราเขียนมันอยู่ในรูปนั้น
เราก็คิดได้ว่า a กับอัตราส่วนร่วมเป็นเท่าใด
แล้วลองเขียนมันเป็นอนุกรมเรขาคณิตจริงๆ ดู
ผมแนะนำให้คุณหยุดวิดีโอนี้
แล้วลองทำด้วยตนเอง
ลองดู อย่างแรกที่คุณอาจสังเกต
คือเรามี 1 ตรงนี้แทนที่จะเป็น 3
งั้นลองแยก 3 ออกมา
อันนี้จึงเท่ากับ 1 ส่วน 3 
คูณ 1 บวก x กำลังสองส่วน 3
และตอนนี้ เนื่องจาก
เราไม่อยากได้ 3 ในตัวส่วน

English: 
So as we talked about
in the last video,
we've seen many
examples of starting
with a geometric
series expanded out,
and then assuming
that its common ratio,
that the absolute value of the
common ratio is less than 1,
finding what the sum
of that might be.
We've proven with this
formula in previous videos.
But now let's go the
other way around.
Let's try to take
some function-- let's
say h of x being equal to
1 over 3 plus x squared--
and let's try to
put it in this form.
And then once we
put it in that form,
we can think about what a
and our common ratio is.
And then try to represent it
as an actual geometric series.
So I encourage you
to pause the video
and try to do that right now.
So let's see, the first
thing that you might notice
is we have a 1 here
instead of a 3.
So let's try to factor out a 3.
So this is equal to 1 over 3
times 1 plus x squared over 3.
And now, since we don't want
that 3 in the denominator,

Portuguese: 
Como falamos no último vídeo,
nós vimos vários exemplos
começando com uma
série geométrica expandida,
e assumindo que a sua razão,
que o valor absoluto da razão
seja menor que um,
queremos encontrar o 
resultado dessa soma.
Nós o provamos com esta
fórmula em vídeos anteriores.
Agora façamos ao contrário.
Vamos tentar tomar 
uma função
-- digamos h de x igual a 1 sobre 3 
mais x ao quadrado --
e tentemos colocar isso nesta forma.
E uma vez que fizermos isso,
poderemos pensar em o 
que a e a nossa razão será.
E tentar representar isso 
como uma série geométrica.
Então, eu lhe encorajo 
a pausar este vídeo
e tentar fazer isso agora.
Então vejamos, a primeira coisa 
que você irá perceber
é que nós temos um 1 
aqui ao invés de um 3.
Vamos tentar isolar um 3.
Isso é igual a 1 sobre 3 vezes
1 mais x ao quadrado sobre 3.
E agora, como não queremos 
esse 3 no denominador,

Korean: 
이것을 1/3으로 생각할 수 있습니다
그래서 이제는 1/3가 분자가 되고
보라색으로 바꿔볼게요
분모에는 1
그리고 우리는 덧셈 하지 않고
공비를 뺄셈 해야 합니다
그래서 1 빼기 그리고 공비를 여기다 적고
 
1-(-x^2/3)
저 식을 이런 꼴로 나타내었으니까
우리는 그 총합이..
이쪽에다가 쓸게요
새로운 색깔로 쓸게요
파란색으로요
n=0부터 무한대번째 항까지의 총합은
초항이 1/3이니까
1/3 곱하기 공비의 n제곱

Thai: 
เราก็คิดว่ามันเป็น 1 ส่วน 3 ได้
เราบอกได้ว่า นี่คือ 1/3 ส่วน --
ขอผมใช้สีม่วงนะ
1/3 ส่วน 1
และเราไม่อยากบวก
เราอยากลบอัตราส่วนร่วม
1 ลบ -- ขอผมเขียนอัตราส่วนร่วมของเราตรงนี้
ด้วยสีเหลือง
1 ลบลบ x กำลังสองส่วน 3
ตอนนี้เราเขียนมันในรูปนั้นได้
ตอนนี้เราบอกได้ว่าผลบวก -- 
ขอผมเขียนมันตรงนี้
-- ขอผมใช้สีใหม่นะ
ขอผมเขียนด้วยสีใหม่
ตอนนี้เราบอกได้ว่า ผลบวกจาก n
เท่ากับ 0 ถึงอนันต์ของ -- 
ลองดู เทอมแรกคือ 1/3
1/3 คูณอัตราส่วนร่วมยกกำลัง n

English: 
we can think about
this as 1 over 3.
So we could say this
is 1/3 over-- let
me do it in that purple color.
1/3 over 1.
And we don't want to
just add something,
we want to subtract
our common ratio.
So 1 minus-- and let me
write our common ratio here
in yellow.
1 minus negative
x squared over 3.
So now we've written
this in that form.
And so now we could say that
the sum-- let me write it here
in-- let me do it
in a new color.
So let me do it in blue.
So now we could say
that the sum from n
equals 0 to infinity of-- let's
see, our first term is 1/3.
1/3 times our common
ratio to the n-th power.

Bulgarian: 
можем да представим 
това като 1/3.
Можем да кажем, че 
това е 1/3 върху...
ще го направя с виолетово.
1/3 върху 1.
И не искаме просто
да прибавим нещо,
искаме да извадим частното.
Значи 1 минус... ще напиша
частното тук в жълто.
1 минус –х^2 върху 3.
Сега представихме това
в този вид.
И сега можем да кажем, 
че сумата... ще я напиша тук
с някакъв нов цвят.
Ще използвам синьо.
Можем да кажем, че
тази сума за n от 0 до безкрайност...
да видим, първият член е 1/3.
1/3 по частното на n-та степен.

Portuguese: 
podemos ver isso como 1 sobre 3.
Podemos dizer então
que 1 sobre 3 sobre
-- deixe-me fazer 
isso em roxo --
1/3 sobre 1.
E não queremos 
somente somar algo,
nós queremos subtrair 
a nossa razão.
1 menos -- vou escrever a 
razão aqui em amarelo.
1 menos x negativo 
ao quadrado sobre 3.
Agora escrevemos 
isso nesta forma.
E agora podemos dizer que a soma
-- deixe-me escrever isso aqui --
vou escrever com uma nova cor.
Escreverei em azul.
Agora podemos dizer 
que a soma de n
igual a 0 ao infinito de -- vejamos,
nosso primeiro termo é 1 sobre 3.
1/3 vezes a nossa razão 
elevada à n-ésima potência.

Bulgarian: 
Частното е –х^2 върху 3.
Ако искаме да развием това,
това ще бъде равно на...
първият член е 1/3 по
всичко това на нулева степен.
Значи става 1/3.
Всеки следващ член е просто
предходния член по частното.
Значи 1/3 по (–х^2)/3
става –1/9 по х^2.
Това трябва да умножим по...
1/3 по –1/3,
умножаваме по –1/3.
И умножаваме по х^2.
Следващият член – умножаваме
отново по –(х^2)/3.
Това става плюс –
минус по минус е плюс,
+1/27 по х^4.
х^2 по х^2 става
х на четвърта степен.

Thai: 
อัตราส่วนร่วมคือลบ x กำลังสองส่วน 3
และถ้าเราอยากกระจายมันออกมา
มันจะเท่ากับ -- เทอมแรกคือ 1/3 คูณ
ทั้งหมดนี้ยกกำลัง 0
มันก็แค่ 1/3
แล้วเทอมต่อไปก็คือ
เทอมที่แล้วคูณอัตราส่วนร่วมของเรา
1/3 คูณลบ x กำลังสองส่วน 3
จะเท่ากับลบ 1/9 x กำลังสอง
จากตรงนั้นถึงตรงนั้น คุณ
ต้องคูณด้วย -- ลองดู 1/3, ลบ 1/3
คุณต้องคูณมันด้วยลบ 1/3
แล้วเราคูณด้วย x กำลังเช่นกัน
ตอนนี้ เทอมต่อไปของเรา เราจะ
คูณด้วยลบ x กำลังสองส่วน 3 อีก
มันจะเป็นบวก -- ลบคูณลบ
ได้บวก -- บวก 1/27 x กำลังสี่
x กำลังสองคูณ x กำลังสอง ได้ x กำลังสี่

Korean: 
공비는 (-x^2/3)
그리고 이것을 전개하면
그래서 첫번째 항은 1/3 곱히기
이거 전체의 0제곱이 됩니다
그래서 그냥 1/3이 되죠
그 다음항들은 그냥
이전 항에 공비를 곱한 것이 되겠죠
1/3 곱히기 (-x^2/3)는
1/9 x^2가 됩니다
이제 여기서 여기로 왔고
뭐를 곱하냐면.. 1/3 에서 -1/3
이제는 -1/3을 곱해야합니다
그리고 x^2까지 말입니다
그리고 그다음 항은
다시  (-x^2/3)를 곱합니다
이것은 +가 되겠네요. - 곱하기 - 는
+니까 1/27 x^4가 됩니다
x 제곱에 x 제곱을 곱하면 x의 4제곱이 되죠

Portuguese: 
A razão é x negativo ao 
quadrado sobre 3.
E se quiséssemos expandir isso,
isso seria igual a -- o primeiro
termo será 1/3 vezes
tudo isso elevado a zero.
Isso então será simplesmente 
igual a 1 sobre 3.
Então cada termo 
sucessivo será somente
o termo anterior vezes a razão.
Logo, 1/3 vezes x negativo 
ao quadrado sobre 3
será 1/9 vezes x 
negativo ao quadrado.
Para ir disso para 
aquilo, você tem que
multiplicar por -- vejamos,
1/3 para 1/3 negativo,
você tem que multiplicá-lo
por 1/3 negativo.
E nós multiplicamos por 
x ao quadrado também.
Agora, no nosso próximo 
termo, nós iremos
multiplicar por x ao 
quadrado sobre 3 de novo.
Logo isso será -- a 
negativo vezes a negativo
será a positivo -- mais 1/27 
vezes x elevado à quarta.
x ao quadrado vezes x 
ao quadrado, x à quarta.

English: 
Common ratio is negative
x squared over 3.
And if we wanted
to expand this out,
this would be equal to-- so
the first term is 1/3 times
all of this to the 0-th power.
So it's just going to be 1/3.
And so each successive
term is just
going to be the previous
term times our common ratio.
So 1/3 times negative
x squared over 3
is going to be
negative 1/9 x squared.
To go from that
to that, you have
to multiply by-- let's
see, 1/3 to negative 1/3,
you have to multiply
it by negative 1/3.
And we multiplied by
x squared as well.
Now in our next
term, we're going
to multiply by negative
x squared over 3 again.
So it's going to be plus-- a
negative times a negative is
a positive-- plus
1/27 x to the fourth.
x squared times x squared,
x to the fourth power.

Portuguese: 
E continuamos dessa maneira.
E quando convergir, no 
intervalo de convergência,
isso irá convergir para h de x.
Qual será o intervalo 
de convergência aqui?
E eu lhe encorajo a pausar 
o vídeo e pensar nisso.
Bom, o intervalo 
de convergência
é o intervalo sobre 
o qual a razão,
o valor absoluto da razão, 
é menor que 1.
Eu vou escrever isso aqui.
O valor absoluto de x negativo 
ao quadrado sobre 3
tem que ser menor que um.
Bom, o valor absoluto será
um número negativo.
Isso é o mesmo 
que dizer que --
vou rolar isso um pouco.
Isso é o mesmo que dizer 
que o valor absoluto
de x ao quadrado sobre 3 
tem que ser menor que 1.

Thai: 
และเราก็ทำต่อไปเรื่อยๆ ได้
และเมื่อตัวนี้ลู่เข้า บนช่วงการลู่เข้า
อนุกรมนี้จะลู่เข้าหา h ของ x
แล้วช่วงการลู่เข้าตรงนี้คืออะไร?
ผมแนะนำให้คุณหยุดวิดีโอแล้วลองคิดดู
ช่วงการลู่เข้า
คือช่วงที่อัตราส่วนร่วมของคุณ
ค่าสัมบูรณ์ของอัตราส่วนร่วมน้อยกว่า 1
ขอผมเขียนค่านี้ลงไปนะ
ค่าสัมบูรณ์ขอองลบ x กำลังสองส่วน 3
ต้องน้อยกว่า 1
ค่าสัมบูรณ์ นี่คือ
ค่าลบ
อันนี้เหมือนกับบอกว่า --
ขอผมเลื่อนลงมาหน่อย
นี่ก็เหมือนกับบอกว่า ค่าสัมบูรณ์ของ x
กำลังสองส่วน 3 ต้องน้อยกว่า 1

English: 
And we just keep going
on and on and on.
And when this converges, so over
the interval of convergence,
this is going to
converge to h of x.
Now, what is the interval
of convergence here?
And I encourage you to pause
the video and think about it.
Well, the interval
of convergence
is the interval over
which your common ratio,
the absolute value of your
common ratio, is less than 1.
So let me write this
right over here.
So our absolute value of
negative x squared over 3
has to be less than 1.
Well, the absolute
value, this is
going to be a negative number.
This is the same
thing as saying--
let me scroll down a little bit.
This is the same thing as saying
that the absolute value of x
squared over 3 has
to be less than 1.

Korean: 
이제 이것을 계속해서 하면
이게 수렴 반경 안에서 수렴할 때
이 식은 h(x)에 수렴하게 됩니다
그럼 여기에서 수렴 반경은 무엇일까요
잠시 동영상을 멈추고 
스스로 생각해 보는 것을 추천드립니다
수렴반경은
공비의 절댓값이
1보다 작은 구간입니다
이 쪽에다 써 볼게요
(-x^2/3) 의 절대값은
1보다 작아야 합니다
이 식은 음수이므로
절대값은
이것은 이것과 같습니다
잠시만 아래로 내릴게요
이것은
|x^2/3|가 1보다 작다는 것과 같습니다

Bulgarian: 
И продължаваме така
до безкрай.
Когато това е сходящо, когато
сме в интервала на сходимост,
това е сходящо към h(х).
Какъв е интервалът 
на сходимост?
Насърчавам те да спреш
видеото и да помислиш.
Интервалът на сходимост е
интервалът, в който частното,
абсолютната стойност на
частното е по-малка от 1.
Ще го запиша тук.
Абсолютната стойност на
(–х^2)/3 е по-малка от 1.
Абсолютната стойност, това
ще бъде отрицателно число.
Това е същото, като да кажем...
ще превъртя малко надолу.
Това е същото като да кажем, 
че абсолютната стойност
на (–х^2)/3 трябва да е 
по-малка от 1.

Thai: 
และนี่ก็เหมือนกับบอกว่า --
อย่างหนึ่งที่คุณอาจสะดุดตา
คือว่า x กำลังสอง
อันนี้จะเป็นบวกไม่ว่ายังไงก็ตาม
หรือจะบอกว่า พจน์นี้
ไม่เป็นลบ ไม่ว่ายังไงก็ตาม
นี่ก็เหมือนกับบอกว่า x กำลังสองส่วน 3
ต้องน้อยว่า 1
จริงไหม?
ผมไม่อยากให้คุณงงในขั้นตอนนี้
ค่าสัมบูรณ์ของ x กำลังสองส่วน 3
จะเท่ากับ x กำลังสองส่วน 3 เพราะตัวนี้ไม่มี
ทางมีค่าลบ
แล้วเราก็คูณทั้งสองข้างด้วย 3
ผมจะขึ้นไปทำตรงนี้นะ
คูณทั้งสองข้างด้วย 3 บอกว่า x กำลังสอง
ต้องน้อยกว่า 3
แล้วนั่นหมายความว่า ค่าสัมบูรณ์ของ x
ต้องน้อยกว่ารากที่สองของ 3
หรือเราบอกได้ว่า x มากกว่า
ลบรากที่สองของ 3
และมันน้อยกว่ารากที่สองของ 3
นี่ก็คือช่วงการลู่เข้า

Korean: 
그리고 이것은 또한
여기서 잠깐 눈치채셨을 것이
x^2은 항상 양수라는 것입니다
아니죠 이렇게 말하는 것이 올바를 것입니다
항상 음수가 아니라는 것이죠
그래서 이것은 결국
x^2/3가 1보다 작다는 것과 같습니다
이해 되시나요?
이 부분에서 여러분이 햇갈리면 안됩니다
|x^2/3|는
그냥  x^2/3가 됩니다.
이 부분은 절대 음수값이 되지 못하기 때문입니다
양변을 3으로 곱하면
이쪽으로 올라와서 다시 쓰면
양변을 3으로 곱해
x^2가 3보다 작아야 합니다
결국 |x|의 값이
√3보다 작아야 합니다
x는
-√3보다 크고
√3보다는 작다는 것이죠
그럼 이것이 수렴 반경입니다

English: 
And this is another
way of saying--
well, one thing that might jump
out at you is that x squared,
this is going to be
positive no matter what.
Or I guess I should
say, this is going
to be non-negative
no matter what.
So this is another way of
saying that x squared over 3
has to be less than 1.
Right?
I don't want to confuse you
in this step right over here.
But the absolute value
of x squared over 3
is just going to be x squared
over 3, because this is never
going to take on
a negative value.
And so we can multiply
both sides by 3.
I'll go up here now to do it.
Multiply both sides by
3 to say that x squared
needs to be less than 3.
And so that means that
the absolute value of x
needs to be less than
the square root of 3.
Or we could say
that x is greater
than the negative
square root of 3,
and it is less than
the square root of 3.
So this is the interval
of convergence.

Bulgarian: 
Това е друг начин да кажем, че...
нещо, което може би ти хрумва,
е че х^2
ще бъде положително
при всички положения.
Предполагам, че мога
да кажа, че това
ще бъде винаги 
неотрицателно.
Това е друг начин да кажем,
че (–х^2)/3 трябва да е по-малко от 1.
Нали?
Не искам да те обърквам
в тази стъпка.
Но абсолютната стойност
на (–х^2)/3
е просто (–х^2)/3, защото това 
никога не може да е отрицателно.
Можем да умножим 
двете страни по 3.
Ще го направя тук горе.
Умножаваме двете страни по 3,
тогава х^2 трябва
да е по-малко от 3.
Това означава, че
абсолютната стойност на х
трябва да е по-малко от
квадратен корен от 3.
Можем да кажем, че
х е по-голямо от
отрицателен корен 
квадратен от 3,
и е по-малко от 
квадратен корен от 3.
Това е интервалът
на сходимост.

Portuguese: 
E isso é outra 
maneira de dizer --
uma coisa que pode chamar a 
sua atenção é que x ao quadrado,
isso será positivo 
de qualquer forma.
Ou acho que eu deveria dizer,
isso será não-negativo 
de qualquer forma.
Então isso é uma outra 
maneira de dizer
que x ao quadrado sobre 
3 tem que ser menor que 1.
Certo? Eu não quero confundir 
você nesse passo.
Mas o valor absoluto de x 
ao quadrado sobre 3
será somente x ao 
quadrado sobre 3,
porque isso nunca 
terá um valor negativo.
E nós podemos multiplicar 
ambos os lados por 3.
Irei fazer isso aqui em cima.
Multiplique os dois lados por 3 para
mostrar que x ao quadrado 
tem que ser menor que 3.
Então isso significa que o 
valor absoluto de x
precisa ser menor que 
a raiz quadrada de 3.
Ou podemos dizer 
que x é maior
que o negativo da 
raiz quadrada de 3,
e é menor que a 
raiz quadrada de 3.
Este é o intervalo de convergência.

Thai: 
นี่คือช่วงการลู่เข้าของอนุกรมนี้
ของอนุกรมกำลังนี้
มันคืออนุกรมเรขาคณิต ซึ่ง
ก็คือกรณีพิเศษของอนุกรมกำลัง
และตลอดช่วงการลู่เข้านี้
มันจะเท่ากับ 1 ส่วน 3 บวก x กำลังสอง
ตราบใดที่ x อยู่ในช่วงนี้
มันจะมีค่าเดียวกัน
กับฟังก์ชันเดิมของเรา เป็นแนวคิดที่เจ๋งดี
 

Bulgarian: 
Това е интервалът на сходимост
за този степенен ред.
Това е геометричен ред, който
е частен случай на
степенните редове.
И в интервала на сходимост
това ще бъде равно на
1/(3 + х^2).
Когато х принадлежи
на този интервал,
той ще има същите стойности
като оригиналната функция,
което е много елегантна идея.

Portuguese: 
Esse é o intervalo de 
convergência dessa série,
desta série de potências.
É uma série geométrica, que é
um case especial de 
uma série de potência.
E neste intervalo de convergência,
isso será igual a 1 sobre 3 
mais x ao quadrado.
Logo, enquanto x 
estiver neste intervalo,
ele terá os mesmos valores
da nossa função original, 
o que é bem interessante.
Legendado por 
[Musa Morena Marcusso Manhães]

Korean: 
이 급수의 수렴 반경이죠
이것은 기하급수로
멱급수의 특수한 형태이죠
수렴 반경 내에서
그 값은
1/(x^2/3)가 됩니다
x가 이 범위 안에 있는 이상은
이 값은 항상
우리의 원래 식과 같은 값이 될 것입니다
상당히 재미있는 일이죠
 

English: 
This is the interval of
convergence for this series,
for this power series.
It's a geometric
series, which is
a special case of
a power series.
And over the interval
of convergence,
that is going to be equal
to 1 over 3 plus x squared.
So as long as x is
in this interval,
it's going to take
on the same values
as our original function,
which is a pretty neat idea.
