Sequences and series theorems

Question 1

triangle inequality

Answer 1

|a+b| ≤ |a| + |b|

for all a,b∈ℝ

Question 2

uniqueness of limits

Answer 2

if a sequence (aₙ) converges then its limit l is unique

Question 3

Algebra of limits

Answer 3

suppose (aₙ) and (bₙ) are sequences converging to l and m respectively. Then:

(i) (aₙ) + (bₙ) -> l + m
(ii) q(aₙ) -> ql for all q∈ℝ
(iii) (aₙ)(bₙ) -> lm
(iv) (aₙ)/(bₙ) -> l/m provided m!=0 l,m∈ℝ

Question 4

Sandwhich theorem

Answer 4

Let N∈ℕ and l∈ℝ. suppose we have (aₙ)<=(bₙ)<=(cₙ) for all n>N
if (aₙ)->l and (cₙ)->l then (bₙ) must converge to l

Question 5

convergence of subsequences

Answer 5

suppose (aₙ)->l (l∈ℝ) and aₙₖ is a subsequence then aₙₖ->l

Question 6

Bolzano-wierstrass theorem

Answer 6

every bounded sequence of real numbers includes a convergent sequence

Question 7

Monotone convergence theorem

Answer 7

(i) if (aₙ) is increasing and bounded above then it converges
(ii) if (aₙ) is decreasing and bounded below then it converges

Question 8

Eulers theorem

Answer 8

The sequence (aₙ) given (aₙ) = (1 + 1/n)ⁿ converges to e

Question 9

sum of geometric series

Answer 9

S = (1-rᴺ⁺¹)/(1-r)

Question 10

sum to infinity of a geometric series

Answer 10

S = 1/(1-r)

Question 11

Algebra of limits for series theorem

Answer 11

suppose Σ∞ₙ₌₁(aₙ) and Σ∞ₙ₌₁(bₙ) converge and l,m∈ℝ then Σ∞ₙ₌₁l(aₙ)+m(bₙ) converges, Σ∞ₙ₌₁(laₙ+mbₙ) = lΣ∞ₙ₌₁(aₙ)+mΣ∞ₙ₌₁(bₙ)