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
The sum of 2 divergent sums
may be finite
∑(u[k] + v[k]) both converge
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 -> ∞
∑ (-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 |
f(x) = f(0) + xf'(0) + ½ x² f''(0) + 1/3! x³ f'''(0) + ...
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
1 + x + ½ x² + 1/3! x³ + ...
x - 1/3! x³ + 1/5! x⁵ - ...
1 - ½ x² + 1/4! x⁴ - ...
(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)