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Prove that for all real a,b

sqrt(a*a + b*b) <= |a| + |b|

sqrt(a*a + b*b) <= |a| + |b|

______

Prove that for all real a,b

1/sqrt(2) * (|a| + |b|) <= sqrt(a*a + b*b)

1/sqrt(2) * (|a| + |b|) <= sqrt(a*a + b*b)

=

Investigate the limit

lim xy / (x*x + y*y) as (x,y) approaches (0,0).

lim xy / (x*x + y*y) as (x,y) approaches (0,0).

doesn't exist

Investigate the limit

lim x*y*y / (x*x + y*y) as (x,y) approaches (0,0).

lim x*y*y / (x*x + y*y) as (x,y) approaches (0,0).

= 0

hint: |xy/(x^2+y^2)| is bounded

hint: |xy/(x^2+y^2)| is bounded

Investigate the limit

lim exp(-x*x-y*y) as (x,y) approaches (oo,oo)

lim exp(-x*x-y*y) as (x,y) approaches (oo,oo)

= 0

Investigate the convergence of the sequence {n^p * z^n}n where
z is a complex number and p is a real number.

if |z|>1, then diverges

if |z|<1, then converges

if |z|=1, then it depends on p, see page 15 in OLDTIMER for inspiration

if |z|<1, then converges

if |z|=1, then it depends on p, see page 15 in OLDTIMER for inspiration

State and prove Weierstrass Test for Unifrom Convergence of
Series.

(1) /\n:-|N /\x:-A | f[n](x) | <= a[n]

(2) the series a[n] converges

thesis:

(3) the series f[n](x) converges uniformly on A

(4) the series | f[n](x) | converges uniformly on A

(2) the series a[n] converges

thesis:

(3) the series f[n](x) converges uniformly on A

(4) the series | f[n](x) | converges uniformly on A

Let R be the radius of convergence of a power series with
center at 0.

Prove that the series converges uniformly on

{z: |z| <= r} for all r, 0<r<R.

Prove that the series converges uniformly on

{z: |z| <= r} for all r, 0<r<R.

Use Weierstrass Test.

very important hint: r<q<R.

very important hint: r<q<R.

Find the radius of converges of the series ((-1)^n / n) *
z^(n(n+1)).

R=1

hint: a[n(n+1)] = (-1)^n / n

hint: a[n(n+1)] = (-1)^n / n

Let a[n], b[n] be sequences of complex numbers, indexed from
zero.

Suppose that series |a[n]| and series b[n] both converge.

What can we infer from this? (State without proof.)

Suppose that series |a[n]| and series b[n] both converge.

What can we infer from this? (State without proof.)

Let /\n:-|N C[n] = +(k=0 to n) [ a[n-k]*b[k] ] = +(k=0 to n) [
a[k]*b[n-k] ].

Let A = +(n=0 to oo) a[n].

Let B = +(n=0 to oo) b[n].

Then +(n=0 to oo) C[n] = A * B

Let A = +(n=0 to oo) a[n].

Let B = +(n=0 to oo) b[n].

Then +(n=0 to oo) C[n] = A * B