Suppose (zn) is convergent to z and z0

1. Homework 5

If z, w are two complex numbers, we use zw or z · w to denote the multiplication of complex numbers z ? w defined in class.

(1) Let (z_n) be a sequence of complex numbers. Suppose (z_n) is convergent to z and z⁰. Show that z = z⁰.

Proof. Use triangle inequality:

|z − z⁰| ≤ |z_n− z| + |z_n− z⁰|.

 (2) Let (z_n) and (w_n) be sequences of complex numbers. Suppose that (z_n) and (w_n) are

convergent to z and w respectively for some z, w ∈ C.

(a) Show that (z_n± w_n) is convergent to z ± w.

(b) Show that (z_nw_n) is convergent to zw.

(c) Show that if w 6= 0, then there exists N ∈ N such that wⁿ 6= 0 for all n ≥ N. In this case, show that (zn/wn)n≥N is convergent to z/w.

Proof. These proofs are the same as those in R. For (a), use triangle inequality:

|z_n± w_n− (z ± w)| ≤ |z_n− z| + |w_n− w.|

For (b), use

|znwn− zw| ≤ |zn− z||wn| + |z||wn− w|.

For (c), we may use (b). You need to show that kwnk is nonzero when n sufficiently large.

 (3) Suppose r, ρ > 0. z = r(cos θ + i sin θ) and w = ρ(cos φ + i sin φ) where 0 ≤ θ, φ < 2π. Prove

the following identities:

(a) zw = rρ(cos(θ + φ) + i sin(θ + φ));

(b) 1 w = 1

ρ(cos(−φ) + i sin(−φ));

(c) zⁿ= rⁿ(cos nθ + i sin nθ), for any n ∈ Z.

Proof. For (a), use basic trigonometric identities.

For (b), you may use (a).

For (c), use induction and (b).

 (4) Let r > 0 and z = r(cos θ + i sin θ) with 0 ≤ θ < 2π.

(a) Sum the geometric progression

1 + z + z²+ · · · + zⁿ,

take the real and imaginary parts and so obtain compact formulas for r sin θ + r²sin 2θ + · · · + rⁿsin nθ

and for

1

2 + r cos θ + r²cos 2θ + · · · + rⁿcos nθ.

Use 1 + z + z²+ · · · + zⁿ= 1 − zⁿ⁺¹ 1 − z .

(b) Let sn = 1 + z + z²+ · · · zⁿ, for n ≥ 1. Show that if r = |z| < 1, then

n→∞lim sn= 1 1 − z.

1

2

Proof. This is proved in class. 

(c) Use (b) to obtain formulas for 1

2 +

∞

X

n=1

rⁿcos nθ and

∞

X

n=1

rⁿsin nθ.

if 0 < r < 1.

(d) Use (c) to prove that for 0 < r < 1, 1 − r

1 + r ≤1 2 +

∞

X

n=1

rⁿcos nθ ≤1 + r 1 − r.

(5) Test the convergence/divergence of the infinite series of complex numbers

∞

X

n=1

iⁿ n²− i. (6) Let a₁, · · · , a_n be real numbers. Show that

√1 n

n

X

j=1

|aj| ≤q

a²₁+ · · · + a²_n ≤

n

X

j=1

|aj|.

(7) On R³, we define the norm (length) of a vector v = (v₁, v₂, v₃) by kvk =

q

v²₁+ v₂²+ v₃². We define the inner product of v, w ∈ R³by

hv, wi = v1w1+ v2w2+ v3w3. Show that

(a) ka · vk = |a|kvk.

(b) hv, vi = kvk².

(c) kvk = 0 if and only if v = 0.

(d) |hv, wi| ≤ kvkkwk.

(e) kv + wk ≤ kvk + kwk.

Proof. This is proved in class.

 (8) We say that a sequence of vectors (vn) is convergent to v in R³if for any  > 0, there exists

N∈ N such that

kv_n− vk < , whenever n ≥ N_.

We say that a sequence (vn) of vectors in R³ is a Cauchy sequence in R³ if for any  > 0, there exists N∈ N such that

kv_n− v_mk < , whenever n, m ≥ N_. Suppose vn= (xn, yn, zn) ∈ R³ for all n ≥ 1.

(a) Show that (vn) is convergent to v = (x, y, z) if and only if

n→∞lim x_n= x, lim

n→∞y_n= y, lim

n→∞z_n= z.

Proof. This is proved in class.

 (b) Suppose (vn) is convergent. Show that (vn) is a Cauchy sequence in R³.

Proof. This is proved in class.

 (c) Show that (vn) is a Cauchy sequence in R³ if and only if (xn), and (yn), and (zn) are

Cauchy sequences in R.

3

Proof. This is proved in class.

 (d) Use (c) to prove that R³ is complete, i.e. every Cauchy sequence in R³ is convergent.

Proof. This is proved in class.

 (e) Evaluate the limit of the sequence of vectors (v_n) in R³defined by

v_n=



n tan⁻¹ 1

n, 3n + 1 3n − 1

n

, n² 2n − 1sin1

n



, n ≥ 1

(9) A sequence of vectors (vn) in R³ is bounded in R³ if there exists M > 0 such that kvnk ≤ M for all n ≥ 1.

Suppose vn= (xn, yn, zn) for n ≥ 1.

(a) Show that a convergent sequence of vectors in R³ is always bounded.

Proof. This is proved in class.

 (b) Show that if (v_n) is bounded in R³, then (x_n) and (y_n) and (z_n) are all bounded in R.

Proof. This is proved in class.

 (c) Use (b) to prove the Bolzano-Weierstrass theorem: any bounded sequence of vectors in

R³ has a convergent subsequence in R³.

(10) Let 0 < C < 1. Suppose that (vn) is a sequence of vectors in R³ and v0 be a vector in R³ such that

kvn+1− vnk ≤ Ckvn− vn−1k, n ≥ 1.

Show that (vn) is a convergent sequence of vectors in R³. Proof. Show that (vn) is a Cauchy sequence in R³.

 (11) Let (vn) be a sequence of vectors in R³.

(a) Show that if

∞

X

n=1

kvnk is convergent in R, then

∞

X

n=1

vn is convergent in R³.

Proof. This is proved in class.

 (b) Let v_n = (−1)ⁿ⁻¹

n

 1

√3, 1

√3, 1

√3



for n ≥ 1. Show that

∞

X

n=1

v_n is convergent in R³

while

∞

X

n=1

kvnk is divergent in R.

Proof. Since kv_nk = 1/n, P∞

n=1kvnk is divergent by p-test. While

∞

X

n=1

vn = 1

√3

∞

X

n=1

(−1)ⁿ⁻¹

n ,

∞

X

n=1

(−1)ⁿ⁻¹

n ,

∞

X

n=1

(−1)ⁿ⁻¹ n

!

is convergent by Leibnitz test (for each component). 

4

(c) Suppose that there exists a sequence of positive real numbers (bn) such that 0 ≤ kvnk ≤ bn, n ≥ 1.

Show that if

∞

X

n=1

b_n is convergent in R, then

∞

X

n=1

v_n is convergent in R³. Proof. This is proved in class.

 (d) Prove that if

∞

X

n=1

vn is convergent in R³, then lim

n→∞vn= 0. Here 0 = (0, 0, 0).

Proof. This is proved in class.

 (12) (Ratio/Root Test for infinite series of vectors in R³.) Let (vn) be a sequence of vectors in

R³and

ρ = lim

n→∞

kvn+1k

kvnk or = lim

n→∞

pkvn nk.

Prove the following statement.

(a) If ρ < 1,

∞

X

n=1

vn is convergent in R³.

(b) If ρ > 1,

∞

X

n=1

vn is divergent in R³. Proof. This is proved in class.