Series

Question 1

What is a series and a partial sum?

Answer 1

S_n = Σ_i=0ⁿ a_i is the nth partial sum of the series Σ_i=0^∞a_i
The initial term does not need to be indexed as the 0th term

Question 2

What is a Cauchy sequence?

Answer 2

A sequence of real numbers S₁, S₂, … is called a Cauchy sequence (or fundamental sequence) if for any ε > 0 there exists N₀ such that for all i, j > N₀, |S_i - S_j| < ε

Question 3

What is the relevance of Cauchy sequences to convergence?

Answer 3

A theorem states that a sequence of real numbers converges iff it is a Cauchy sequence

Question 4

How do you find the formula for a geometric series?

Answer 4

S_n+1 = S_n + acⁿ⁺¹ = cS_n + a, rearranging the two expressions gives the required formula when c ≠ 1

Question 5

What is the nth degree Taylor polynomial of f around the point a?

Answer 5

P_n(x) = f(a) + f’(a)(x - a) + f’‘(a)(x - a)²/2! + … + f⁽ⁿ⁾(a)(x - a)ⁿ/n!

Question 6

How can the Taylor polynomial be described?

Answer 6

Since P_n^(k)(a) = f^(k)(a) for each k = 0, …, n, P_n has the same ‘local information’ as f

Question 7

What is the Taylor series?

Answer 7

When f has derivatives of all orders at x = a, its Taylor series about the point x = a is Σ_k=0^∞ f^(k)(a)(x - a)^k/k!

Question 8

When is a Taylor polynomial a good approximation of a function at a point?

Answer 8

When the Taylor series at x converges to f(x)
Taylor polynomials do not provide a good approximation of f at x if the series diverges for some x or the series converges for x but to a value other than f(x)

Question 9

How can the linear approximation be derived?

Answer 9

L(x) = n + mx should have an error term f(x) - L(x) which goes to zero faster than x - a as x –> a, formally lim_{x –> a} (f(x) - L(x))/(x - a) = 0
This implies f(a) = L(a) so f(a) = n + ma and f is differentiable at a with f’(a) = L’(a) = m
Thus L(x) = f(a) + f’(a)(x - a)
Informally, f(x) ≈ f(a) + f’(a)(x - a)
Higher order Taylor polynomials are finding an error term which goes to zero faster than (x - a)ⁿ as x –> a

Question 10

What is Taylor’s Remainder Theorem?

Answer 10

If f is n times continuously differentiable on [a, x], and n + 1 times differentiable on (a, x), then the error term R_n(x) = f(x) - P_n(x) = f⁽ⁿ⁺¹⁾(c)(x - a)ⁿ⁺¹/(n + 1)! for some c ∈ [a, x]
If x < a, replace x and a in the bounds