Math 521 Midterm 3

definition of convergence of an infinite series

if s_n=∑_k=1ⁿ(a_k) [sequence of partial sums] converges to s, then ∑_k=1^∞(a_k) [infinite series] converges to s

the divergence test

if lim(a_k)≠0 then ∑_k=1^∞(a_k) diverges (note: not IFF; consider harmonic series ∑(1/k))

series convergence with boundedness

a series with nonnegative terms converges iff the sequence of partial sums (s_n) is bounded

basic comparison test

Let N₀∈N; 1) Suppose |a_n|≤c_n ∀n≥N₀ then if ∑(c_k) converges, ∑|a_k| and ∑(a_k) converge; 2) Suppose a_n≥d_n≥0 ∀n≥N₀ then if ∑(d_k) diverges, ∑(a_k) diverges

Algebraic Limit Theorem for infinite series

1) if ∑a_k converges and c∈R then ∑ca_k=c∑a_k; 2) if ∑a_k converges and ∑b_k converges then ∑(a_k+b_k) = ∑a_k + ∑b_k

Limit Comparison Test

Suppose x_n and y_n are positive sequences and lim(x_n/y_n)=r for r≥0 exists. Then, 1) if r>0 then ∑x_k converges IFF ∑y_k converges; 2) if r=0 then if ∑y_k converges then ∑x_k converges

absolute convergence def

∑a_k converges absolutely if ∑|a_k| converges

conditionally convergent

∑a_k converges but NOT absolutely

re-arrangement def

∑b_k is a re-arrangement of ∑a_k if there exists a bijection f:N→N s.t. ∀k b_k=a_f(k)

Root Test

Let a_k be a sequence of nonnegative real numbers. Suppose lim(ⁿ√a_n)=r. Then, r<1 implies ∑a_k converges; r>1 implies ∑a_k diverges; r=1 inconclusive

When to use which test?

ALWAYS start with divergence; for powers use root test; for factorials use ratio test; for all else use BCT

Ratio Test

Let a_k be a sequence of positive real numbers. Suppose [(a_n+1)/(a_n)] converges to λ for some λ∈[0,∞). Then, λ<1 implies ∑a_k converges; λ>1 implies ∑a_k diverges; λ=1 is inconclusive

monotone convergence theorem

if a sequence is monotone and bounded then it converges

alternating series test

let a_n≥0 and suppose a₁≥a₂≥a₃… and lim a_n as n→∞ =0 then ∑_k=1^∞(-1)^k(a_k) converges

def of a limit

Let A⊆R, c a lim pt of A and f: A→R. Then L is the limit of f at c if ∀ε>0 ∃δ>0 s.t. ∀x∈A
0<|x-c|<δ → |f(x)-L|<ε

Let A⊆R, c a lim pt of A and f: A→R. Then TFAE

lim as x→c of (f(x))=L; ∀(x_n)_n=1^∞ ∈A s.t. x_n≠c ∀n and x_n→c then lim as n→∞ (f(x_n))=L

continuous at c def

Let A⊆R, f:A→R and c∈A. We say f is continuous at c if ∀ε>0 ∃δ>0 s.t. 0<|x-c|<δ → |f(x)-f(c)|<ε

Intermediate Value Theorem

Let f:[a,b]→R be continuous on [a,b]. Suppose f(a)<k<f(b) or f(b)<k<f(a). Then ∃c∈[a,b] s.t. f(c)=k

def of f’(x)

f’(x)=lim as t→x (f(t)-f(x))/(t-x)=lim as h→0 (f(x+h)-f(x))/h

differentiable and continuous theorem

if f:(a,b)→R is differentiable at c∈(a,b) then f is continuous at c

chain rule

Suppose f:(a,b)→R and g:I→R (I is an interval) with f((a,b))⊆I. If f is differentiable at c∈(a,b) and g is differentiable at f(c)∈I then (g°f)’(c)=g’(f(c))(f’(c))

Rolle’s theorem

Suppose f:[a,b]→R is continuous. if f is differentiable on (a,b) and f(a)=f(b) then ∃c∈(a,b) s.t. f’(c)=0

generalized MVT

Suppose f:[a,b]→R and g:[a,b]→R are continuous on [a,b] and differentiable on (a,b) then ∃c∈(a,b) s.t. [f(b)-f(a)]g’(c)=[g(b)-g(a)]f’(c)

L’Hopital’s Rule

Suppose I is an interval, f and g are continuous and differentiable on I except possibly at x=a∈I. Suppose f(a)=g(a)=0, g’(x)≠0 if x≠a. Then lim as x→a of (f’(x)/g’(x))=L implies lim as x→a of (f(x)/g(x))=L

geometric series

let r∈R then ∑_n=0^∞rⁿ converges IFF |r|<1 and converges to 1/(1-r)

absolute convergence theorem

if ∑|a_n| converges then ∑a_n converges

Extreme Value Theorem

Let K⊆R be nonempty and compact and let f:K→R be continuous, then ∃x_min and x_max s.t. ∀x∈K, f(x_min)<=f(x)<=f(x_max)

uniform continuous def

f is uniformly continuous on A if ∀ε>0 ∃δ>0 s.t. ∀x,y ∈ A, |x-y|<δ→|f(x)-f(y)|<ε

preservation of compactness

Let K⊆R be compact and f:K→R be continuous, then f(K) is compact

cauchy criterion/ sequence convergence theorem

to determine if something converges without knowing its limit; ∑(ak) converges IFF ∀ε>0 ∃N∈N s.t. ∀n≥m≥N |∑(ak)|<ε

bolanzo-weirstrass theorem

every bounded sequence in R has a convergent subsequence

Uniform continuity theorem

if K⊆R is compact and f:K→R is continuous then f is uniformly continuous on K

Darboux’s theorem

if f is differentiable on an interval then f’ has the IVT property

compactness in R

K⊆R is compact IFF K is closed and bounded

MVT

Let f:[a,b]→R be continuous on [a,b] and differentiable on (a,b), then ∃c∈(a,b) s.t. f’(c)=(f(b)-f(a))/(b-a)

def of a limit (2022 q1)

c is a limit point of S iff ∃(z_n)⊆S\{c} s.t. z_n→c