Series
Introduced| English | Meaning |
|---|---|
| converges/kənˈvɜːdʒɪz/ | Approaches a finite limit as the relevant variable or sequence index increases or approaches a specified value. |
| geometric series/ˌdʒiːəʊˈmetrɪk ˈsɪəriːz/ | geometric series |
| binomial expansion/baɪˈnəʊmɪəl ekˈspænʃn/ | The expression obtained by expanding a power of a sum of two terms. |
| binomial coefficient/baɪˈnəʊmɪəl ˌkəʊɪˈfɪʃənt/ | The number of ways to choose a specified number of items from a set, also appearing in a binomial expansion. |
| arithmetic progression/əˈrɪθmətɪk prəˈɡreʃn/ | A sequence in which the difference between consecutive terms is constant. |
| common difference/ˈkɒmən ˈdɪfrəns/ | The constant amount added to each term to obtain the next term in an arithmetic sequence. |
| geometric progression/ˌdʒiːəʊˈmetrɪk prəˈɡreʃn/ | A sequence formed by multiplying each term by the same constant to obtain the next. |
| common ratio/ˈkɒmən ˈreɪʃɪəʊ/ | The factor multiplying each term to obtain the next term in a geometric sequence. |
| sum to infinity/sʌm tʊ ɪnˈfɪnɪti/ | The limiting value of the partial sums of a convergent infinite series. |
The infinite sum that converges 收敛
- Add $1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \cdots$ forever. The total? Exactly $2$.
- This is a geometric series 等比级数 with ratio $\dfrac{1}{2}$. Not all infinite sums behave this nicely — but the ones that do are powerful tools in finance, physics, and computing.
The binomial expansion 二项展开式
- For a positive integer $n$:
- where $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$ is a binomial coefficient 二项式系数.
Worked example. $(1+x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4$. The coefficients $1, 4, 6, 4, 1$ are the 4th row of Pascal's triangle.
What is the binomial coefficient C(5, 2) = 5! / (2! 3!)?
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C(5,2) = 120/(2×6) = 120/12 = 10.
Arithmetic progressions 等差数列 (AP)
- An AP adds a constant common difference 公差 $d$ each step: $a, a+d, a+2d, \ldots$
- $n$th term: $u_n = a + (n-1)d$.
- Sum of first $n$ terms: $S_n = \dfrac{n}{2}\big(2a + (n-1)d\big)$.
An AP has first term a = 3 and common difference d = 4. What is the 5th term?
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u₅ = a + (n−1)d = 3 + 4×4 = 3 + 16 = 19.
An AP has a = 2 and d = 3. What is the sum of the first 5 terms?
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S₅ = (5/2)(2×2 + 4×3) = (5/2)(4 + 12) = (5/2)(16) = 40.
Geometric progressions 等比数列 (GP)
- A GP multiplies by a constant common ratio 公比 $r$ each step: $a, ar, ar^2, \ldots$
- $n$th term: $u_n = ar^{n-1}$.
- Sum of first $n$ terms: $S_n = \dfrac{a(1-r^n)}{1-r}$ (for $r \neq 1$).
AP grows linearly (add $d$); GP grows exponentially (multiply by $r$). For $r > 1$, the GP eventually dwarfs the AP.
Build a sequence
uₙ = a · rⁿ⁻¹
Each term multiplies by r. When |r| < 1 the terms shrink and the sum to infinity converges.
A GP has a = 2 and r = 3. What is the 4th term?
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u₄ = ar³ = 2 × 3³ = 2 × 27 = 54.
Sum to infinity 无穷和
- A GP converges when $|r| < 1$ — the terms get smaller and smaller.
- Sum to infinity: $S_\infty = \dfrac{a}{1-r}$.
$|r| < 1$ is essential. If $|r| \geq 1$, the terms don't shrink and the infinite sum diverges (grows without bound). The formula $S_\infty = \dfrac{a}{1-r}$ only applies when the series converges.
- An arithmetic progression has a common difference; a geometric progression a common ratio. A convergent geometric progression ($|r|<1$) shows convergence to a sum to infinity.
A GP has a = 8 and r = 0.5. What is the sum to infinity?
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S∞ = a/(1−r) = 8/(1−0.5) = 8/0.5 = 16.
A geometric series with r = 1.5 has a finite sum to infinity.
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The sum to infinity formula only applies when |r| < 1. With r = 1.5, the series diverges.
Match each formula to what it computes.
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d marks the arithmetic family, r the geometric one — recognising the formula is half the exam question.
You've got it
- binomial expansion: $(a+b)^n = \sum \binom{n}{r} a^{n-r} b^r$
- AP: $u_n = a + (n-1)d$, $S_n = \dfrac{n}{2}(2a+(n-1)d)$
- GP: $u_n = ar^{n-1}$, $S_n = \dfrac{a(1-r^n)}{1-r}$
- if $|r|<1$, sum to infinity $S_\infty = \dfrac{a}{1-r}$