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Test the series 1/n * sin(pi/4 * n) for convergence.

converges, Dirichlet-Abel test

Test the series (-1)^n / (n - (-1)^n) for convergence.

The series converges.

Define S[n] = +(k=1 to n) (-1)^k / (k - (-1)^k).

Prove that lim(n->oo) S[2n] = log(2).

Prove that lim(n->oo) S[2n+1] = log(2), too.

See page 83 in OLDTIMER about this log(2).

The number log(2) is not essential here.

Define S[n] = +(k=1 to n) (-1)^k / (k - (-1)^k).

Prove that lim(n->oo) S[2n] = log(2).

Prove that lim(n->oo) S[2n+1] = log(2), too.

See page 83 in OLDTIMER about this log(2).

The number log(2) is not essential here.

The series a(n) converges.

Does the series a(n)*a(n) have to converge?

Does the series a(n)*a(n) have to converge?

NO.

a(n) = (-1)^n / sqrt(n)

a(n) = (-1)^n / sqrt(n)

Let a(n) be a sequence of positive numbers.

Suppose that the series a(n) converges.

Does the series a(n)*a(n) have to converge?

Suppose that the series a(n) converges.

Does the series a(n)*a(n) have to converge?

Yes.

Let a(n) be a sequence of positive numbers.

Suppose that the series a(n) converges.

Does the series n*a(n) / (n+a(n)) have to converge?

Suppose that the series a(n) converges.

Does the series n*a(n) / (n+a(n)) have to converge?

Yes.

n/(n+a(n)) is less than 1

n/(n+a(n)) is less than 1

Let f be a convex function defined on an interval on the real
line. Does it have to be continuous?

NO.

f:[0,1] -> R

f(1) = 1

f(x) = 0

This function is convex but is not continuous.

f:[0,1] -> R

f(1) = 1

f(x) = 0

This function is convex but is not continuous.

What is a contraction?

(X,d) metric space

f : X -> X

f is a contraction iff there exists 0 < K < 1 such that for every x,y belonging to X we have d( f(x), f(y) ) <= K * d(x,y).

f : X -> X

f is a contraction iff there exists 0 < K < 1 such that for every x,y belonging to X we have d( f(x), f(y) ) <= K * d(x,y).

State and prove Banach's Contraction Principle.

In a complete metric space every contraction has a unique fixed
point.

(x is a fixed point <=> f(x) = x)

Also known as Banach's Fixed Point Theorem.

page 56 in golden gate

(x is a fixed point <=> f(x) = x)

Also known as Banach's Fixed Point Theorem.

page 56 in golden gate

Prove that in a complete metric space, a countable intersection
of a collection of open dense sets is dense.

page 133 in golden gate

Baire Category Theorem

Baire Category Theorem

Prove that if a complete metric space is written as a union of
countably many closed sets, then at least one of those closed
sets contains a ball.

Use the fact that in a complete metric space, a countable
intersection of a collection of open dense sets is dense.

page 133 in golden gate

page 133 in golden gate