# NST IA Maths: Infinite Series

## 37 terms

### Infinite Series

Summations of infinite numbers of terms

### The infinite series converges (to S)

if the partial sums have a finite limit (S) as n -> ∞

### Infinite series that do not converge

may diverge (± ∞) or oscillate

may be finite

A + B

### u[k] -> 0 as k -> ∞

is a necessary (but not sufficient) condition for convergence

### A series is absolutely-convergent

if ∑[0,∞) |Uk| converges

### Absolutely Convergent Series

Have the same sum regardless of the order

### Conditionally convergent series

Can be made to sum to any value or diverge, by rearranging terms

### Geometric Progression, Sn = ∑ rⁿ

(1 - rⁿ⁺¹) / (1 - r)

### Geometric Progression S∞ = ∑ rⁿ

1 / (1 - r) where | r | < 1 otherwise diverges

### Harmonic Series ∑ (1 / k) = 1 + ½ + 1/3 + ...

Divergent (shown by grouping terms)

### Comparison Test - Converges

if uk ≤ vk for k ≥ some K and ∑v converges

### Comparison Test - Diverges

if uk ≥ vk for k ≥ some K and ∑v diverges

### Ratio Test - Converges

lim u[k+1] / uk < 1 as k -> ∞

### Ratio Test - Diverges

lim u[k+1] / uk > 1 as k -> ∞

### Ratio Test - may converge

lim u[k+1] / uk = 1 as k -> ∞

### Leibnitz Criterion

∑ (-1)ⁿ⁺¹ an converges (providing ak > 0 and is monotonic decreasing and as k -> ∞, ak -> 0)

### Power Series, f(x)

∑an xⁿ = a0 + a1 x + a2 x² + ...

### Power Series - Converges (absolutely)

|x| < 1 / L (where L is the lim | a[k+1] / ak | )

### Power Series - Diverges

|x| > 1 / L (where L is the lim | a[k+1] / ak | )

### Power Series - may converge

|x| = 1 / L (where L is the lim | a[k+1] / ak | )

### Complex Power Series

(absolute) convergence of |z| < 1 / lim | a[k+1] / ak |

### Maclaurin Series

f(x) = f(0) + xf'(0) + ½ x² f''(0) + 1/3! x³ f'''(0) + ...

### Taylor Series

f(x) = f(a) + (x - a) f'(a) + ½ (x - a)² f''(a) +1/3! (x - a)³ f'''(a) + ...

### Remainder Term Rn

1 / (n - 1)! ∫[x,0] (x-t)ⁿ⁻¹ fⁿ(t) dt

### Taylor Series must converge (to f(x))

if lim as n -> ∞ of Rn = 0

### exp(x)

1 + x + ½ x² + 1/3! x³ + ...

### sin x

x - 1/3! x³ + 1/5! x⁵ - ...

### cos x

1 - ½ x² + 1/4! x⁴ - ...

### Binomial Theorem

(1 + x)ⁿ = 1 + nx + ½n(n-1) x² + n(n-1)(n-2) / 3! x³ + ...

### ln(1 + x)

x - ½ x² + 1/3 x³ - 1/4 x⁴ + ...

### Newton - Raphson Method

x₁ = x₀ - f(x₀) / f'(x₀)

### Newton - Raphson Method - errors

εi+₁ ≈ ei² f''(x) / 2f'(x)

Diverges

Converges

Converges