4. series : Σ notation

수열의 합

series

" Σ 기호를 사용하니까

합 표시가 너무 편리해요"

" it's very compact to use Σ notation "

수열은 순서대로 나열된 숫자들의 공통된 규칙을 찾아내는 마치 게임과도 같은 재미있는 단원입니다.

영리한 학생들은 초등산수 시절부터 나열된 항 사이의 계차로 쉽게 그 규칙성을 찾아내기도 하지만, 부분분수나 군수열 등 조금 더 어려운 유형들을 해결하려면 기본적인 유형들에 대한 어느 정도의 해법 암기도 필요합니다.

이번에는 수열의 정의와 사용되는 기호 등의 가장 기초가 되는 내용들을 설명하려고 합니다. 첫 단계부터 기초 개념과 원리을 확실하게 이해해 두기 바랍니다.

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지난번에, 짝수들의 수열 2, 4, 6, 8, 10, ... 의 제 n 번째 항은 a_n = 2n 이라고 표현한다는 것을 배웠습니다.

We learned that the n-th general term of the sequence of even numbers 2, 4, 6, 8, 10, ... is a_n = 2n.

이제, 수열들의 제 n 번째 항까지의 합인 2 + 4 + 6 + ... + 2n 은 기호로 S_n 이라고 나타냅니다.

The summation of this sequence, 2 + 4 + 6 + ... + 2n, up to the n-th term is denoted by S_n.

일반화해서 문자로 나타낸다면, 무한개의 항을 갖는 수열 { a_n } 에 대하여 아래와 같이 표현할 수 있습니다.

As for an infinite sequence { a_n }, we use general notations as follows :

{ a_n } = a₁, a₂, a₃, ... ,a_n, ...

S_n = a₁ + a₂ + ... + a_n = \(\sum\limits_{k = 1}^n {{a_k}} \)

S = a₁ + a₂ + ... + a_n + ... = \(\sum\limits_{k = 1}^\infty  {{a_k}} \)

그러면 위와 같이, 수열의 합을 간단하게 표현할 때 아주 유용하게 쓰이는 Σ (씨그마, sigma) 기호에 대하여 알아 보도록 합니다.

Now, let me introduce you a symbol Σ (sigma) which means to sum up the sequence terms as shown above.

예를 들어, 1 + 2 + 3 + ... + 100 과 같이 일정한 규칙으로 더해지는 수열의 합은 Σ 기호를 사용해서 아주 간단하게 표현할 수 있습니다.

For example, we can use sigma notation Σ to express the summation of 1 + 2 + 3 + ... + 100.

1 + 2 + 3 + ... + 100 = \(\sum\limits_{k = 1}^{100} k \)

(1) Σ 기호의 아래 부분에는 위의 예에서의 k = 1 과 같이, 변수 k 와 시작하는 값 1 을 작은 글씨로 적습니다

In the lower part of a symbol Σ, write an index k and the starting value 1 as k = 1 shown above.

(2) Σ 기호의 윗 부분에는 위의 예와 같이, 변수 k 가 끝나는 값 100 을 작은 글씨로 적습니다. 만일 무한수열이라면 ∞ 로 표시하면 됩니다.

On the upper part of a notation Σ, write the ending value 100 of an index k and we use symbol ∞ for infinite sequences.

(3) Σ 기호의 오른쪽에는 더해야 하는 식이나 값을, 변수k 로 나타내서 보통 크기로 적습니다. 이 때, 변수는 i, n, p ... 등의 다른 문자를 사용해도 됩니다.

Now, on the right hand side of a symbol Σ, write what to sum in terms of an index k as shown above. Also, you can use i, n, p ... for an index of summation.

Σ 기호의 사용에 익숙해질 수 있도록, 보기 문제들을 풀어 보도록 할까요?

Let's try some worked examples in order to be accustomed to using Σ notations.

(1) \(\sum\limits_{k = 2}^{17} {{k^2}} \) = 2² + 3² + ... + 17²

(2) \(\sum\limits_{n = 1}^{15} {3n} \) = (3 x 1) + (3 x 2) + ... + (3 x 15)

(3) \(\sum\limits_{i = 1}^{20} 2 \) = 2 + 2 + 2 + ... + 2 = 2 x 20 = 40

이번에는 반대로, 수열의 합으로 표시된 것을 Σ 기호로 나타내 보도록 할까요?

On the contrary, now let's change the following sum of sequences into Σ notations.

(1) 1 + 4 + 7 + 10 + ... + 61 = \(\sum\limits_{n = 1}^{21} {(3n - 2)} \)

(2) 2 + 4 + 8 + 16 + ... = \(\sum\limits_{k = 1}^\infty  {{2^k}} \)

(3) 1 x 2 + 3 x 4 + ... + 19 x 20 = \(\sum\limits_{p = 1}^{19} {p(p + 1)} \)

이제 Σ 기호의 성질에 대하여 공부해 보도록 합니다.

Let's investigate the properties of summation notation Σ.

(1) \(\sum\limits_{k = 1}^{20} {(k + {k^2})} \)

= (1 + 1²) + (2 + 2²) +(3 + 3²) + ... + (20 + 20²)

= ( 1 + 2 + 3 + ... + 20) + (1² + 2² + 3² + ... + 20²)

= \(\sum\limits_{k = 1}^{20} k \) + \(\sum\limits_{k = 1}^{20} {{k^2}} \)

(2) \(\sum\limits_{k = 1}^{20} {3k} \)

= 3 x 1 + 3 x 2 + 3 x 3 + ... + 3 x 20

= 3 x ( 1 + 2 + 3 + ... + 20)

= 3\(\sum\limits_{k = 1}^{20} k \)

(3) \(\sum\limits_{k = 1}^n 3 \)

= 3 + 3 + 3 + ... + 3

= 3n

위에서 살펴보았던 기호 Σ 의 성질을 일반화해서 아래와 같이 정리해 두도록 할까요?

We can summarize the general properties of Σ as follows :

---

(1) \(\sum\limits_{k = 1}^n {({a_k} + {b_k})}  = \sum\limits_{k = 1}^n {{a_k}}  + \sum\limits_{k = 1}^n {{b_k}} \)

(2) \(\sum\limits_{k = 1}^n {(p \times {a_k})}  = p \times \sum\limits_{k = 1}^n {{a_k}} \)

(3) \(\sum\limits_{k = 1}^n c  = c \times n\)

---

따라서, 이러한 Σ 의 성질을 이용하면, 복잡한 수열의 합도 아래와 같이 간단하게 계산해 낼 수가 있습니다.

It will be much easier to calculate the sum of complicated sequences if we use the properties of summation Σ as shown below.

\(\sum\limits_{k = 1}^n {(2{k^2} - 3k + 4)}  = 2\sum\limits_{k = 1}^n {{k^2}}  - 3\sum\limits_{k = 1}^n k  + 4n\)

참고로 제 n 항까지의 (1) 자연수, (2) 자연수의 제곱 및 (3) 자연수의 세제곱들의 합의 공식은 아래와 같습니다. 이를 구하는 원리는 다음에 따로 설명하도록 합니다.

For your reference, the sums of (1) positive integers, (2) their squares and (3) cubes up to n-th term are shown below. The ways to find these sums will be explained later.

\(\sum\limits_{k = 1}^n k  = \frac{{n(n + 1)}}{2}\)

\(\sum\limits_{k = 1}^n {{k^2}}  = \frac{{n(n + 1)(2n + 1)}}{6}\)

\(\sum\limits_{k = 1}^n {{k^3}}  = {\left\{ {\frac{{n(n + 1)}}{2}} \right\}^2}\)

마지막으로, 수열의 합 S_n 과 일반항 a_n 의 관계에 대하여 알아 보도록 할까요?

Lastly, let's find out the relationship between the sum S_n and the general term a_n.

S_n = a₁ + a₂ + ... + a_n -1 + a_n ... ①

S_n -1 = a₁ + a₂ + ... + a_n -1      ... ②

이제, 연립방정식과 같이 윗식에서 아래식을 빼주면,

Now, use the similar method to the systems of equations,

[가감법]  ① – ② :

[ elimination method ]

S_n – S_n -1 = a_n ( n ≥ 2 )

이 식은 앞으로 배우게 될, 여러가지 수열의 점화식 등에서 활용되는 매우 중요한 일반항 유도 공식이니까, 반드시 기억해 두기 바랍니다.

This result will be very useful when we learn various types of recursive formulas later.