Partial fraction decomposition (部分分数分解)

d = 0, e = 1 ⇒ Eq. 1/((fx + p)(gx + q) = a/(fx + p) + b/(gx + q)

2x² + x - 4 = a(x + 1)(x + 2) + bx(x + 2) + cx(x + 1)

= (a + b + c)x² + (3a + 2b + c)x + 2a

∴ a + b + c = 2, 3a + 2b + c = 1, 2a = -4 → (a, b, c) = (-2, 3, 1)
∴ (2x² + x - 4)/(x(x + 1)(x + 2)) = 2/x + 3/(x + 1) + 1/(x + 2)

5x³ - 12x² + 6x + 17 = (ax + b)(x² + 4x + 7) + (cx + d)(2x² -4x + 3)
∴ a +2c = 5, 4a +b -4c +2d = -12, 7a +4b +3c -4d = 6, 7b +3d = 17
→ (a, b, c, d) = (-1, 2, 3, 1)
∴ (5x³ - 12x² + 6x + 17)((2x² - 4x + 3)(x² + 4x + 7))

= (-x + 2)/(2x² - 4x + 3) + (3x + 1)/(x² + 4x + 7)

1 = a(x + 1)² + bx(x + 1) + cx = (a + b)x² + (2a + b + c)x + a
∴ a + b = 0, 2a + b + c = 0, a = 1 → (a, b, c) = (1, -1, -1)
∴ 1/(x(x + 1)²) = 1/x -1/(x + 1) -1/(x + 1)²

|x| ≥ 0, |-x| = |x|, |x|² = x², |xy| = |x||y|, -|x| ≤ x ≤ |x|

(a). x - 4 ≥ 0 ⇔ x ≥ 4 & [x - 4 < 3x → x → x > -2] ⇒ x ≥ 4
(b). x - 4 < 0 ⇔ x < 4 & [-(x - 4) < 3x → x > 1] ⇒ 1 < x < 4
⇒ x > 1

(a) x < 7 ⇒ (x - 7 < 0 & x - 8 < 0) ⇒ -(x - 7) -(x - 8) < 3

6 < x < 7

(b) 7 ≤ x < 8 ⇒ (x - 7 ≥ 0 & x - 8 < 0) ⇒ +(x - 7) -(x - 8) < 3

(⇒ 1 < 3 ⇒ all ℜ) 7 ≤ x < 8

(c) 8 ≤ x ⇒ (x - 7 ≥ 0 & x - 8 ≥ 0) ⇒ +(x - 7) +(x - 8) < 3

8 ≤ x < 9

⇔ 6 < x < 9

∴ –(|a| + |b|) ≤ a + b ≤ |a| + |b| → |a + b| ≤ |a| + |b| // = Q.E.D. (L)

^∀a > 0, ^∀b > 0 ⇒ (a + b)/2 ≥ √(ab), having equality (=) when a = b

= (a² + 2ab + b²)/4 - ab = (a² - 2ab - b²)/4 = (a - b)²/4 ≥ 0
when a = b ⇒ (left side) = (right side) //

Sum formula (和の公式)

= (n(n + 1)·(2n + 1))/6 + 2·(n(n + 1))/2 + n = n·(2n² + 9n + 13)/6

= n·(n - 1)!/((k - 1)!{n - 1) - (k - 1)}!) = n_n-1C_k-1 //
(2) _n-1C_k + _n-1C_k-1 = (n - 1)!/(k!(n - 1 - k)!) + (n - 1)!/(k - 1)!(n - k)!
= ((n - 1)·(n - 1)!)/(k!(n - k)!) + (k·(n - 1)!)/(k!(n - k)!)
= (n·(n - 1)!)/(k!(n - k)!) = _nC_k //

(2) _nC₀ - _nC₁ + _nC₂ - … + (-1)ⁿ_nC_n = 0

0!! = 1!! = 1, (n + 2)!! = n!!(n + 2)
even integer only: (2n)!! = 2 × 4 × … × 2n ⇔
odd integer only: (2n – 1)!! = 1 × 3 × … × (2n – 1) [(-1)!! = 0! = 1]

(2n)!! = 2 × 4 × … × 2n = (2 × 1) × (2 × 2) × … × (2 × n) = 2ⁿ × n!
(2n – 1)!! × (2n)!! = [1 × 3 × … × (2n – 1)] × (2 × 4 × … × 2n)

= 1 × 2 × 3 × 4 × … × 2n = (2n)! //

{a_n} = {a₁, a₂, a₃, … a_n}
term, element or member (項): each number in the sequence

first term = a₁__ith term = a_i

Ex. S_n+1 = 2S_n → S_n ≡ general term (一般項)

n_p ∈ N, {a_p} strictly increasing sequence ⇒ {a_np}

{a_2n} = {a₂, a₄, a₆, a₈, …}, {a_n²} = {a₁, a₄, a₉, a₁₆, …}

Δf(x) = f(x + 1) – f(x), Δ⁰f(x):= f(x), Δ²f(x): = Δ (Δf(x)), …

Δ(delta): forward difference operator (前進差分演算子)

∇f(x) = f(x) – f(x – 1), ∇⁰f(x):= f(x), ∇²f(x): = ∇(∇f(x)), …

∇ (nabla): backward difference operator (後進差分演算子)

Def. common difference (公差) ≡ a_n+1 - a_n = d (constant)

Ex. {2, 4, 6, …, 100}

∝ direct propotion (正比例) ↔ ∝^-1 inverse proportion (反比例)

Def. a ≡ first term (初項)
Def. common ratio (公比) ≡ a_n+1/a_n = r (constant)
Ex. {1, 2, 4, 8, 16, …}

Fibonacci sequence (フィボナッチ数列)

a₁ = 0, a₂ = 1, … a_k = a_k-1 + a_k-2 (k > 1)

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …

= general term of Fibonacci sequence
a_n = (1/√5)·[{(1 + √5)/2}ⁿ – {(1 – √5)/2}ⁿ]
__ = (φⁿ - (1 - φ)ⁿ)/√5 = (φⁿ - (-φ)^-n)/√5

Circular constant, π (円周率)

4/π = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + …

π²/6 = 1 + 1/2² + 1/3² + 1/4² + 1/5² + 1/6² + …

ζ function (ゼータ関数), ζ(s)

s = 2 ⇒ Euler's equation

Dirichlet L-function (ディクレルのL関数), L(s)

s = 1 ⇒ Leibniz formula

ε-δ definition (ε-δ論法)

such that n > m implies |x_n - a| < ε

0 < |x – 1| < δ → |3x + 2 – 5| < ε → |3x – 3| < ε → |x – 1| < ε/3

[Take 0 < ε ≤ ε/3]

∵ ^∀ε > 0, ^∃δ < 1/3 → |3x – 3| < ε → |3x + 2 – 5| < ε //

(Solve to be |(n + 1)/(n – 1)| < n₀ to ^∀ε > 0)

→ 2/ε < n – 1, 2/ε + 1 = (2 + ε)/ε + 1 < n
n₀ = (2 + ε)/ε + 1 is determined to (2 + ε)/ε + 1 ≡ n₀ when ^∀ε > 0
|(n + 1)/(n – 1) – 1| = |2/(n – 1)| < ε to n₀ ≤ ^∀n //

From the assumption of proposition, |a - b| < ε₀

= |a - b|/2 ⇒ |a - b| < 0 contradiction //

2. a_n+1/a_n = {Dⁿ⁺¹(n + 1)!}/(Dⁿ/n!)

= D/(n + 1), lim_n→∞|D/(n + 1)| = 0 //

On n ≥ N(ε), |1/n² - 0| = 1/n² ≤ 1/N(ε)² < ε ∴ |1/n² - 0| < ε
From the definition of limit value, lim_n→∞(1/n²) = 0 //

hold when n > 2/ε - 1
0 < ^∀ε < 2, N ∋ N(ε) = [2/ε] ⇒ N(ε) > 2/ε - 1
∴ ^∀n ≥ N(ε) ⇒ |(3n + 5)/(n + 1) - 3| = 2/(n + 1) ≤ 2/(N(ε + 1)) < ε
From the definition of limit value, lim_n→∞((3n + 5)/(n + 1)) = 3 //

lim_n→∞a_n = a ⇒ lim_n→∞|a_n| = |a|
lim_n→∞a_n = 0 ⇔ N.S.C. a ⇒ lim_n→∞|a_n| = 0
lim_n→∞a_n = a ⇔ N.S.C. lim_n→∞|a_n - a| = 0

Basic formula of limits (pl. formulae) (Fr.)

{a_n}, {b_n}, ^∀n ∈ N, a_n ≤ b_n for all n, lim_n→∞a_n = a, lim_n→∞b_n = b

lim_n→∞a_n ≤ lim_n→∞b_n

If lim_n→∞a_n = a, lim_n→∞b_n = b, a > b. ε := (a - b)/2
lim_n→∞a_n = a ⇒ N₁ = N₁((a - b)/2) exist

n ≥ N₁ ⇒ |a_n - a| < (a - b)/2

lim_n→∞b_n = b ⇒ N₂ = N₂((a - b)/2) exist

n ≥ N₂ ⇒ |b_n - b| < (a - b)/2

When n ≥ N₂, |b_n - b| < (a - b)/2 ⇒ b_n - b < (a - b)/2 ⇒ b_n < (a - b)/2
N := max(N₁, N₂), if n ≥ N ⇒ a_n > (a + b)/2 > b_n (contradiction) //

lim_n→∞(a_n ± b_n) = a ± b

(^∃N₂ ∈ N)(^∀n ∈N: n ≥ N₂) |b_n - b| < ε/2
N := max{N₁, N₂}, n ≥ N, n ∈ N →
|(a_n + b_n) - (a + b)| = |(a_n - a) + (b_n - b)|

≤ |a_n - a| + |b_n - b| < ε/2 + ε/2 = ε //

lim_n→∞a_n·b_n = ab

⇒ 0 < ε^* < 1
From the assumption, N₁(ε^*) and N₂(ε^*) exist on ε^*

n ≥ N₁(ε^*) ⇒ |a_n - a| < ε^*, n ≥ N₂(ε^*) ⇒ |b_n - b| < ε^*

N(ε) := max(N₁(ε^*), N₂(ε^*))

a_nb_n - ab = b(a_n - a) + a(b_n - b) + (a_n - a)(b_n - b)

From triangle inequality and 0 < ε^* < 1, if n ≥ N(ε),
|a_nb_n - ab| ≤ |b||a_n - a| + |a||b_n - b| + |a_n - a||b_n - b| <

|b|ε^* + |a|ε^* + ε^*2 = (|a| + |b| + ε^*)ε^* < (|a| + |b| + 1)ε^* = ε //

lim_n→∞(a_n/b_n) = a/b

c ≠ 0, ^∀ε > 0, lim_n→∞a_n = a ⇒ N₁(ε/|c|) exists when ε/|c| > 0
Hold n ≥ N₁(ε/|c|) ⇒ |a_n - a| < ε/|c|
When N(ε) := N₁(ε/|c|), and if n ≥ N(ε) ⇒

|ca_n - ca| = |c(a_n - a)| = |c||a_n - a| < |c|·(ε/|c|) = ε //

= lim_n→∞((1 + 2·0 + 3·0)/(3 + 2·0 + 0)) = 1/3

a_n ≤ c_n ≤ b_n (n ∈ N), lim_n→∞a_n = lim_n→∞b_n = a
⇒ lim_n→∞c_n = a

n ≥ N₁(ε) ⇒ |a_n - a| < ε ∴ a - a_n < ε
n ≥ N₂(ε) ⇒ |b_n - a| < ε ∴ a - b_n < ε
N(ε) := max(N(₁ε), N₂(ε))

n ≥ N(ε) ⇒ a_n > a - ε, b_n < a + ε

If n ≥ N(ε) ⇒ a - ε < a_n ≤ c_n ≤ b_n < a + ε
∴ -ε < c_n - a < ε ⇒ |c_n - a| < ε //

Ax. [I] continuity of real numbers (実数の連続性)

X: bounded

bounded below and unbounded above

Th. [II] Weiestraß theorem

^∀a > 0, ^∀b > 0, n ∈ N ⇒ ^∃na > b

(ii) ^∀x > 0, ^∃n ∈ N, nε > 1 (a := ε, b := 1, axiom of Archimedes) →

ε > 1/n → -(1/n) > 0 - ε ∴ x > 0 - ε (^∃x ∈ C) //

Th. [III]

Accumulate point or cluster point (集積点)

interval (区間)

open interval (開区間) (open set 開集合)

a, b: end point (終点). a, lower limit (下端). b, upper limit (上端)

semi-open interval (半開区間) (semi-closed set 半開集合)

--●-----○→__--○-----●→

closed interval (閉区間), I (closed set 閉集合)

{x|x ≥ a} = [a, ∞)
{x|x > a} = (a, ∞)
{x|x ≤ a} = (-∞, a)
{x|x < a} = (-∞, a]

n = (ⁿ√n)ⁿ = (1 + h_n)ⁿ = Σ_k=0ⁿ_nC_kh_n^k ≥ _nC₂h_n² = (n(n - 1))/2·h_n²
∴ 0 < h_n² ≤ √(2/(n -1)) ⇒ lim_n→∞h_n = 0
∴ lim_n→∞(ⁿ√n) = lim_n→∞(1 + h_n) = 1

Th. [IV] Cantor's intersection theorem (共通部分定理)

Th. [V] Weierstraß-Bolzano (W-B) theorem

(ワイヤシュトラウス-ボルザノの定理)

Ex. formula (公式), e.g., sin²θ + cos²θ = (or ≡) 1

Ex. 5x + 9 = 24
______variable operator
coefficient ↓_____ ↓_constant_constant
_____5___ x_____ +____9___=___24
_____---------__________ --_______---
_____term___________term____ term
_____________left hand side = right hand side

Ex. x/2 = (x + c)/4

x = independent variable ⇔ y = dependent variable

n = 1 ⇒ (x - 1)
n = 2 ⇒ (x - 1)(x + 1)
n = 3 ⇒ (x - 1)(x² + x + 1)
n = 4 ⇒ (x - 1)(x³ + x² + x + 1) = (x - 1)(x + 1)(x² + 1)
n = 5 ⇒ (x - 1)(x⁴ + x³ + x² + x + 1)

= (x - 1)(x + 1)(x² + x + 1)(x² - x + 1)

…
n = n ⇒ (x - 1)(x^n-1 + x^n-2 + … + x + 1)
general form: x = a/b, multiplied by b²

b²{(a/b)² - 1} = b(a/b - 1)·b(a/b + 1)
{b²·(a/b)² - b²} = (b²·a/b - b)·b(b·a/b + b)

xⁿ – 1 = 0_≡_(x – 1)(x^n-1 + x^n-2 + … + x + 1) = 0

Limit (of function) (関数の極限)

|(px + q) - (pa + q)| = |p||x - a| < |p|δ(ε) = ε //

|√(x) - √(a)| = |x - a|/(√(x) + √(a)) ≤ |x - a|/√(a) < δ(ε)/√(a) //

^∀K > 0, δ₁(K) := 1/K > 0, 1 < x < 1 + δ₁(K) ⇒ 1/(x - 1) > 1/δ₁(K) = K
∴ lim_x→1+0(1/(x - 1)) = +∞ (right-hand limit)

^∀K < 0, δ₂(K) := -1/K > 0, 1 - δ₂(K) < x < 1 ⇒ 1/(x - 1) < 1/δ₂(K) = K
∴ lim_x→1-0(1/(x - 1)) = -∞ (left-hand limit)
⇒ right-hand limit (Ex₁.) ≠ left-hand limit (Ex₂.)

^∀ε, δ(ε) > 0, 0 < x < δ(ε) ⇒ 0 < √x < √δ = ε ∴ lim_x→1+0√x = 0

1) limx→af(x) exist.__2) f(a) exist.__3) limx→af(x) = f(a)
⇒ discontinuous when only one of the three conditions is rejected

|f(x) - f(a)| < ε

λf(x) + μg(x), f(x)g(x), f(x)/g(x) (g(x) ≠ 0), |f(x)| continuous

f_n(x) is continuous function at ^∀n, f(x) := lim_n→∞f_n(x)

geometric sequence of first term = x and, common ratio = x

f(x) = limn→∞f_n(x) = limn→∞xⁿ = 0

f(x) = 0 (0 ≤ x < 1), f(x) = 1 (x = 1)

≡ limit of continuous function is not always continuous

lim_x→1-0{lim_n→∞f_n(x)} ≠ lim_n→∞{lim_x→1-0f_n(x)}

f(x) = Σ_n=1^∞f_n(x) = x² + x²/(x² + 1) + x²/(x² + 1)² + …

⇒ f(x) = x²/(1 - 1/(x² + 1)) = x² + 1

f(x) = x² + 1 (x ≠ 0), f(x) = 0 (x = 0)

≡ infinite sum of continuous function is not always continuous

lim_x→0{lim_m→∞Σ_n=1^mf_n(x)} ≠ lim_m→∞{lim_x→0Σ_n=1^mf_n(x)}

lim_x→∞f(x) = ∞__lim_x→-∞f(x) = -∞ ⇒
^∃a, ^∃b (a < b), apply intermediate-value theorem ⇒
a < c < b, f(^∃c) = 0, c = real root //

= limh→0((xⁿ + nx^n-1h + _nC₂x^n-2h² + … + hⁿ) - xⁿ)/h
= limh→0(nx^n-1 + _nC₂x^n-2h + … + h^n-1) = nx^n-1

if f(x) = 1 ⇒ (1/g(x)' = -g'(x)/g(x)²

λ·{f(x + h) - f(x)}/h} + μ·{g(x + h) - g(x)}/h} ⇒ λf'(x) + μg'(x) (h → 0) //

{f(x + h)g(x + h) - f(x)g(x + h) + f(x)g(x + h) - f(x)g(x)}/h =
{f(x + h) - f(x)}/h·g(x + h) + f(x + h)·{g(x + h) - g(x)}/h

⇒ f'(x)g(x) + f(x)g'(x) (h → 0) //

[{f(x + h)g(x) - f(x)g(x + h)}/{g(x + h)g(x)}]/h =
1/{g(x + h)g(x)}·[{f(x + h)g(x) - f(x)g(x) + f(x)g(x) - f(x)g(x + h)}/h] =
1/{g(x + h)g(x)}[{f(x + h) - f(x)}/h}·g(x) - f(x)·{g(x + h) - g(x)}/h]
⇒ 1/{g(x)g(x)}{f'(x)g(x) - f(x)g'(x)} (h → 0)
∴ f(x)/g<x) diff., {f(x)/g<x)}' = {f'(x)g(x) + f(x)g'(x)}/g(x)² //

= 2x(x³ - 2) + (x² + 5)·3x² = 5x⁴ + 15x² - 4x

= 10(x² + 1)⁹·2x = 20x(x² + 1)⁹

∵ y = x^1/m → x^1/m, the relationship of x^m is inverse function

→ (x = y^1/m) y = x^m

∴ dy/dx = 1/ny^m–1 = 1/m·1/(x^1/m)^m–1 = 1/m·x^1/m–1

∴ dy/dx = dy/dt·dt/dx = nt^n-1·(1/m)·x^1/m–1

= n(x^1/m)^n–1·(1/m)·x^1/m–1 = n/m·x^{(n–1)/m+1/m-1} = n/m·x^n/m–1

= 1/h·log_a(1 + h/x) = 1/x … log_a(1 + h/x)^x/h

(∵ lim_x→±∞(1 + 1/x)^x = e)

h → 0, y' → (1/x)·log_ae
∴ dy/dx = log_ae //

= 1/[(1/y)·(log_e1/log_ea)] = y·log_ea = a^x·log_ea

∴ y' = dy/dx = 1/(dx/dy) = 1/(3(³√x²)

Logarithmic differentiation (対数微分法)

(f(x) ± g(x))' = f'(x) ± g'(x), (f(x)·g(x))' = f'(x)·g(x) + f(x)·g'(x)

(F(x + h) – F(x))/h = {f(x + h) ± g(x + h) – (f(x) ± g(x))}/h

= (f(x + h) – f(x))/h ± (g(x + h) – g(x))/h

(G(x + h) – G(x))/h = (f(x + h)·g(x + h) – f(x)·g(x))/h

= (f(x + h) – f(x))/h·g(x) + f(x)·(g(x + h) – g(x))/h

Elementary function ( 初等関数 )

Differentiation of elementary functions (初等関数の微分)

∴ dy/dx = yloga = a^xloga //

(sinx)' = cosx
(cosx)' = -sinx
(tanx)' = (sinx/cosx)' = 1/cos²x = sec²x = 1 + tan²x
(cotx)' = (cosx/sinx)' = -cosec²x
(secx)' = (1/cosx)' = secxtanx
(cosecx)' = (1/sinx)' = -cosecxcotx

(sin^-1x)' = 1/√(1 – x²)
(cos^-1x)' = -1/√(1 – x²)
(tan^-1x)' = 1/(1 + x²)
(cot^-1x)' = -1/(1 + x²)
(sec^-1x)' = 1/(|x|√(x² – 1)
(cosec^-1x)' = -1/(|x|√(x² – 1)

x = siny (-1 < x < 1, -π/2 < y < π/2), x' = cosy > 0
∴ cosy = √(1 – sin²y) = √(1 – x²)
∴ dy/dx = 1/(dx/dy) = 1/cosy = 1/√(1 – x)²

1/2·log(x² + y²)·(x² + y²)'
= 1/(2(x²+ y²)·(2x+ 2y·y') = (x + y·y')/(x²+ y² = (y/x)'/(1 + (y/x)²
= [(x·y' - y)/x²]/(1 + y²/x²) = [(x·y' - y)/x²]/[(x² + y²)/x²]
= (x·y' – y)/(x₂ + y²)
∴ (x + y·y')/(x₂ + y₂) = (x·y' – y)/(x² + y₂) → y' = (x + y)/(x – y) //

(sinhx)' = coshx
(coshx)' = sinhx
(tanhx)' = 1/cosh²x

(1) (e^x)⁽ⁿ⁾ = e^x
(2) (sinx)⁽ⁿ⁾ = sin(x + nπ/2)
(3) (cosx)⁽ⁿ⁾ = cos(x + nπ/2)
(4) {(x + a)^α}⁽ⁿ⁾ = α(α - 1)(α - 2)…(α - n + 1)(x + a)^α-n
(5) {log(x + a)}⁽ⁿ⁾ = (-1)^n-1·(n - 1)!/(x + a)ⁿ (x + a > 0)

(1) {f(ax + b)}⁽ⁿ⁾ = aⁿf⁽ⁿ⁾(ax + b)
(2) {af(x) + bg(x)}⁽ⁿ⁾ = af⁽ⁿ⁾(x) + bg⁽ⁿ⁾(x)

⇒ {f(ax + b)}⁽ⁿ⁾ = aⁿf⁽ⁿ⁾(ax + b) //
(2) trivial because of the linearity of differentiation

n = 1: {f(x)g(x)}' = f'(x)g(x) + f(x)g'(x)
n = 2: {f(x)g(x)}'' = f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)
n = 3: {f(x)g(x)}''' = f'''(x)g(x) + 3f''(x)g'(x) + 3f'(x)g''(x) + f(x)g'''(x)

= x(e^2x)⁽ⁿ⁾ + n(x)'(e^2x)^(n-1) = x·2ⁿe^2x + n·1·2^n-1e^2x = 2^n-1(2x + n)e^2x

Then, express the sum of 5th-order polynomial and residual term.

∴ e^x = Σ_k=0^n-1(1/k!·x^k) + e^θx/n!·xⁿ
∴ e^x = 1 + x + x²/2 + x³/6 + x⁴/24 + x⁵/120 + e^θx/720·x⁶

o(xⁿ)o(x^m) = o(x^n+m)
xⁿo(x^m) = o(x^n+m)
o(xⁿ) ± o(x^m) = o(xⁿ) if m ≥ n
Co(xⁿ) = o(xⁿ), C = constant

e^x = Σ_k=0ⁿ(x^k/k!) + o(xⁿ)
sinx = Σ_k=0ⁿ((-1)^kx^2k+1)/(2k + 1)! + o(x²ⁿ⁺²)
cosx = Σ_k=0ⁿ((-1)^kx^2k/(2k)! + o(x²ⁿ⁺¹)
log(1 + x) = Σ_k=1ⁿ((-1)^k-1x^k)/k + o(xⁿ)
(1 + x)^a = Σ_k=0ⁿ[a:k]x^k + o(xⁿ)

Extreme value (極値)

x = 0, 2 when f'(x) = 0
f''(x) = (2 - 2x)e^-x - (2x - x²)e^-x = (x² - 4x + 2)e^-x
f''(0) = 2 > 0, f''(2) = -2e^-2 < 0
x = 0 → local minimum f(0) = 0, x = 2 → local maximum f(2) = 4/e²

increase rates are different among them

< 2/1·2/n = 4/n → 0 (n → 0)__∴ limn→∞(2ⁿ/n!) = 0 //

n! shows higher increase rate than 2ⁿ

ε = γ - a > 0, N ∋ N = N(γ - a) exist

n ≥ N ⇒ ||a_n+1/a_n| - a| < γ - a

here, if n ≥ N

|a_n+1/a_n| - a ≤ ||a_n+1/a_n| - a| ⇒ |a_n+1| < γ|a_n|

repeat using this inequality when n > N

0 ≤ |a_n| < γ|a_n-1| < γ²|a_n-2| < … < γ^n-N|a_N|

from 0 < γ < 1, limn→∞γ^n-N|a_N| = 0 //

b_n+1/b_n = (n + 1)^a/rⁿ⁺¹·rⁿ/n^a = 1/r·(1 + 1/n)^a ⇒ 1/r (n → ∞)
from 0 < 1/r < 1 and Th. shown above, limn→∞ = 0 //

rⁿ shows higher increase rate than n^a

f'(x) = 1/√(1 - y²)·y' - 1/(1 + x²)

y' = (√(1 + x²) - x·(2x/2√(1 + x²)))/(√(1 + x²) = 1/((1 + x²)(√(1 + x²))
∴ f'(x) = √(1 + x²)·1/((1 + x²)√(1 + x²)) - 1/(1 + x²) = 0
⇒ constant function, f(0) = sin^-10 - tan^-10 = 0 - 0 ∴ f(x) = 0 //

x_n+1 = x_n - (x_n² - a)/2x_n = (x_n² + a)/2x_n

n = 1: (1 + 5)/2 = 3
n = 2: (3² + 5)/(2·3) ≈ 2.333
n = 3: (2.333² + 5)/(2·2.333) ≈ 2.238
n = 4: (2.238² + 5)/(2·2.238) ≈ 2.236

Limit of indeterminate forms (不定形の極限値)

These are not indeterminate forms

seven indeterminate forms which are typically considered
limx→af(x)/g(x) = 0/0, ∞/∞, ∞ – ∞, ∞⁰, 0⁰, or 1^∞
i.e., limit value can not be obtained as it is

∫_a^af(x)dx = 0, ∫_b^af(x)dx = -∫_a^bf(x)dx

lower sum ≡ s_Δ(f) = Σ_k=1ⁿm_k(x_k - x_k-1)
upper sum ≡ S_Δ(f) = Σ_k=1ⁿM_k(x_k - x_k-1)

Σ_k=1ⁿm_k(x_k - x_k-1) ≤ Σ_k=1ⁿf(ξ_k)(x_k - x_k-1) ≤ Σ_k=1ⁿM_k(x_k - x_k-1)
∴ s_Δ(f) ≤ S(f;Δ, {ξk}) ≤ S_Δ(f)

only one k, which supports x_k-1 < c < x_k, exists

Note that S_Δ'(f) ≤ S_Δ(f)

Note that s_Δ(f) ≤ s_Δ'(f)

s_Δ(f) ≤ s_Δ''(f) ≤ S_Δ''(f) ≤ S_Δ'(f) ∴ s_Δ(f) ≤ S_Δ'(f)

s_Δ(f) bounded above on Δ and S_Δ'(f) bounded below on Δ

upper integral ≡ ^-∫_a^bf(x)dx = infΔS_Δ(f) on the greatest lower bound of Δ
lower integral ≡ _-∫_a^bf(x)dx = supΔs_Δ(f) on the least upper bound of Δ

⇒ F(x) – G(x) = C (C: constant of integration 積分定数)
∴ G(x) = F(x) + C = ∫f(x)dx + C = ∫f(x)dx

F(x) = F(a) + (x – a)F'(c) = {f(a) – g(a)} + (x – a)(f'(c) – g'(c))
= (f(a) – (x – a)f'(c)) + (g(a) – (x – a)g'(c)) = f'(a) – g'(a) = k
∴ F(x) = f(x) – g(x) = k //

Integration by substitution (置換積分)

∴ xdx = 1/2dt → I = ∫√t·1/2dt = 1/2·2/3t^3/2 = (1/3)·√(1 + x²)³

[I = ∫(ax + b)/√{(x + p)² + q – p²}dx]
t = x + p →
I = ∫(at + b – ap)/√(t² + q – p²)dt

= ∫at/√(t² + q – p²)dt₍₁₎ + ∫(b – ap)/√(t² + p – q²)dt₍₂₎

(1) I₁: t² + q – p² = u, 2tdt = du

∴ I₁ = a/(2√u)du = a√u = a√(t² + q + p²)

(2) = I₂ = (b – ap)log(t + √(t² + q – p²))
I₁ + I₂ = a√(t² + q + p²) + (b – ap)log(t + √(t² + q – p²))

= a√(x² + 2px + q) + (b – ap)log(x + p + √(x² + 2px + q))

Th. integration by parts (部分積分)

(C, omit – often done)

Basic formulae (基本公式)

Integration of rational functions (有理関数の積分)

∫f(cosx, sinx)dx ⇒ expressed by the integral of rational function on t

∴ ∫1/(2 + cosx)dx = ∫2/(3 + t²)dt

= 2/√3·tan^-1(t/√3) = 2/√3·tan^-1(1/√3·tan(x/2))

∫(cosx/(2 + sin²x))dx

= ∫(1/(t² + 2)dt = 1/√2·tan^-1(t/√2) = 1/√2·tan^-1(sinx/√2)

Integration of irrational function (無理関数の積分)

Basic equations of definite integrals

⇒ ∫_α^βf(x)dx = F(β) - F(α) ≡ [F(x)]_α^β

∴ F(β) - F(α) = (∫_a^βf(t)dt + C) - (∫_a^αf(t)dt + C) =

∫_a^βf(t)dt + ∫_α^af(t)dt = ∫_α^βf(t)dt //

⇒ d/dx∫_a(x)^b(x)f(t)dt = f(b(x))b'(x) - f(a(x))a'(x)

differentiate both sides
d/dx∫_a(x)^b(x)f(t)dt = F'(b(x))b'(x) - F'(a(x))a'(x)

= f(b(x))b'(x) - f(a(x))a'(x) //

Solution of definite integral

2sin(h/2)·SΔ = Σhsin(nh)·2sin(h/2) = h{Σ2sin(nh)·sin(h/2)}

= -h[(cos3h/2 – cosh/2) + (cos5h/2 – cos3h/2)

+ (cos7h/2 – cos5h/2) +
… + {cos(2n + 1)h/2 – cos(2n – 1)h/2}]

∴ 2sin(h/2)·SΔ = -h{cos(2n + 1)h/2 – cos(h/2)},

SΔ = -h{cos(2n + 1)h/2 – cos(h/2)}/{2sin(h/2)}

n → ∞, h = π/n → 0
∴ -h/{2sin(h/2)} → 1, (2n + 1)h/2 = (2n + 1)π/2n → π
∴ limn→∞SΔ = -(cosπ – cos0) = 2

no matter how the relationships between a, b and c are large or small

Δ₁ and Δ₂ are divisions of [a, c] and [c, b], respectively
Set S^[a,b]Δ(f) as interval considering upper sum →
S^[a,b]Δ(f) = Σ_k=1ⁿsupx∈I_kf(x)(x_k - x_k-1)

= (Σ_k=1^l + Σ_k=l+1ⁿ)supx∈I_kf(x)(x_k - x_k-1) = S^[a,c]Δ₁(f) + S^[c,b]Δ₂(f)

Here, |Δ₁| ≤ |Δ|, |Δ₂| ≤ |Δ|, select c as cutoff point, |Δ| < 0

⇒ |Δ₁| < 0, |Δ₂| < 0

^-∫_a^bf(x)dx = ^-∫_a^cf(x)dx + ^-∫_c^bf(x)dx ⇒ true on upper integral

As well, true on lower integral

f(x) integrable on [a, b] ⇒ ^-∫_a^bf(x)dx = _-∫_a^bf(x)dx ⇒

^-∫_a^cf(x)dx + ^-∫_c^bf(x)dx = _-∫_a^cf(x)dx + _-∫_c^bf(x)dx

when ^-∫_a^cf(x)dx = _-∫_a^cf(x)dx, ^-∫_c^bf(x)dx = _-∫_c^bf(x)dx, the equation holds

∴ f(x) integrable on [a, c] //

Here, limt→+0(2 - √t) = 2 (convergence) ∴ ∫₀¹(1/√x)dx = 2

limt→+0(-logt) = ∞ (divergence)

∫_t¹logxdx = [xlogx - x]_t¹ = -1 - tlogt + t. From l'Hôpital's rule,

limt→+0tlogt = limt→+0∫_t¹logt/(1/t) = limt→+0((1/t)/(-1/t²)) = limt→+0(-t) = 0

∫₀¹logxdx = limt→+0∫_t¹logxdx = limt→+0(-1 - tlogt + t) = -1

∴ ∫_-1¹(1/x)dx = ∫_-1⁰(1/x)dx + ∫₀¹(1/x)dx_divergence → divergence

f(x) integrable on [a, ∞], ∫_a^∞|f(x)|dx convergence ⇒
∫_a^∞f(x)dx convergence, as in any other intervals, (a, b], [a, b), (-∞ b]

∫_If(x)dx ≡ conditional converge (条件収束)

Γ-function and β-function (ガンマ関数・ベータ関数)

Area (面積), S

≡ the area enclosed by y = f(x) and lines, x = a and x = b

≡ the area enclosed by y = f(x), y = g(x) and lines, x = a and x = b

x/(x² + 4) - x/8 = (x(4 - x²)/(8(x² + 4)) ≥ 0 ⇔ 0 ≤ x ≤ 2
∴ S = 2∫₀²{ x/(x² + 4) - x/8}dx = [log(x² + 4) - x²/8]₀² = log2 - 1/2

r = sinθ and θ = 0 (0 ≤ θ ≤ π/2)

= 1/2∫₀^π/2(1 + 2cosθ + cos²θ - sin²θ)dθ
= 1/2∫₀^π/2(1 + 2cosθ + cos2θ)dθ = [θ + 2sinθ + sin2θ/2]₀^π/2 = 1 + π/4

Length (長さ), l

C → x = φ(t), y = ψ(t) (α ≤ t ≤ β), φ(t), ψ(t) C¹-class ⇒

L = ∫_α^β√{(dx/dt)² + (dy/dt)²}dt = ∫_α^β√{φ'(t)² + ψ'(t)²}dt

L = ∫_a^b√(1 + f'(x)²)dx

∴ L = ∫₀^log2coshxdx = [sinhx]₀^log2 = sinh(log2) = (2 - 1/2)/2 = 3/4

L = ∫_α^β√(f(θ)² + f'(θ)²)dθ

(dx/dθ)² + (dy/dθ)² = (f'(θ)cosθ - (f(θ)sinθ)² + (f'(θ)sinθ - (f(θ)cosθ)²

= f'(θ)²(cos²θ + sin²θ) + f(θ)²(cos²θ + sin²θ) = f'(θ)² + f(θ)²

∴ L = ∫_α^β√((dx/dθ)² + (dy/dθ)²)dθ = ∫_α^β√(f(θ)² + f'(θ)²)dθ //

Volume and lateral (surface) area (体積・側面積), V and S_l

V = π∫_-a^ay²dx = π∫_-a^a(a² - x²)dx = π/6·{a - (-a)}³ = 4/3·πa³

y' = -x/√(a² - x²) → √(1 + y'²) = √(1 + x²/(a² - x²)) = a/y

∴ S_l = 2π∫_-a^ay√(1 + y'²)dx = 2π∫_-a^aadx = 4πa²

Def. nth item (第n項) = a_n
Def. s_n: nth partial sum (第n部分和) = S_n

the sum of geometric sequence between the first term to nth term when the first term = 1 and common ratio = p
(p = 1 → I_n = n)
I_n = 1 + p + p² + … + pⁿ → pI_n = p + p² + p³ + … + pⁿ⁺¹
∴ I_n - pI_n = (1 – p)I_n = 1 – pⁿ⁺¹ ∴ I_n = (1 – pⁿ⁺¹)/(1 – p)

Cf. Mersenne number (メルセンヌ数)

|x| ≤ 1 → divergence
x = -1 → oscillation (振動) ⊂ divergence

|r| < 1 ⇒ Σ_n=1^∞ar^n-1 = a/(1 - r) convergence
|r| ≥ 1 ⇒ Σ_n=1^∞ar^n-1 divergence

Σ_n=1^∞(aa_n ± bb_n) = aΣ_n=1^∞a_n ± bΣ_n=1^∞b_n

∴ aS + bT (n → ∞) //

nth partial sum, S_n = Σ_k=1ⁿ{1/k - 1/(k + 1)}

= (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + … + (1/n - 1/(n + 1))
= 1 - 1/(n + 1) ∴ convergence

∴ Σ_n=1^∞{1/(n(n + 1))} = limn→∞S_n = limn→∞(1 - 1/(n + 1)) = 1

Sequence of functions (関数列)

Σ_n=1^∞(1/n²) convergence (Weierstrass M-test)
→ Σ_n=1^∞(sinnx/n²) convergence //

a_n := (n + 1)/n! → limn→∞|a_n/a_n+1| = limn→∞[(n + 1)/n!·{(n + 1)!/(n + 2)}]