Show that ∞ X n=1 an

(1)

1. hw 5

(1) Let (an) and (bn) be sequence of real numbers. Assume that there exists N ∈ N so that for any n ≥ N, an≤ bn. Assume further that lim

n→∞an= ∞. Show that lim

n→∞bn= ∞.

(2) Let (an) and (bn) be sequence of real numbers. Assume that 0 < an ≤ bn for all n ≥ 1.

(a) Suppose that both

∞

X

n=1

an and

∞

X

n=1

bn are convergent. Show that

∞

X

n=1

an≤

∞

X

n=1

bn.

(b) Suppose that

∞

X

n=1

a_n= ∞. Show that

∞

X

n=1

b_n= ∞.

(3) Evaluate (a)

∞

X

n=1

1

n(n + 1)(n + 2). (b)

∞

X

n=1

1

n(n + 1)(n + 3).

(4) Let a₁, · · · , a_mbe positive integers with a₁< a₂< · · · < a_m. Define a polynomial Q(x) by Q(x) = (x + a1) · · · (x + am).

Evaluate

∞

X

n=1

1 Q(n).

(5) A sequence (an) of real numbers is a positive sequence if an> 0 for all n ≥ 1. Assume that (a_n) is a positive sequence so that lim

n→∞na_n = c with c > 0. Show that the infinite series

∞

X

n=1

an is divergent.

(6) Let (an) be a sequence of real numbers.

(a) Suppose

∞

X

n=1

|an| is convergent. Show that

∞

X

n=1

a²_n is convergent.

(b) Is the converse of the above statement true? It yes, prove it. If not, provide a coun- terexample.

(7) An infinite series

∞

X

n=1

a_n is called absolutely convergent if

∞

X

n=1

|a_n| is convergent.

(a) Suppose

∞

X

n=1

a²_n is convergent. Show that

∞

X

n=1

a_n

n is absolutely convergent.

(b) Suppose

∞

X

n=1

a²_n and

∞

X

n=1

b²_n are both convergent. Show that

∞

X

n=1

anbn is absolutely convergent.

(8) Prove that

∞

X

n=3

1

n(ln n)^p converges when p > 1 and diverges when p < 1.

(9) Determine whether the series converges or diverges. Explain your answer.

(a)

∞

X

n=1

(−1)ⁿ n n + 1.

1

(2)

2

(b)

∞

X

n=1

sin π n². (c)

∞

X

n=1

n!

nⁿ. (d)

∞

X

n=1

(−1)ⁿ⁻¹ 1 2n + 1. (e)

∞

X

n=1

cos nπ 2ⁿ . (f)

∞

X

n=1

n + 2 (n + 1)³. (g)

∞

X

n=1

(2n − 1)(n²− 1) (n + 1)(n²+ 4)². (h)

∞

X

n=1

cos^nπ₃

√n .

(i)

∞

X

n=1

cos^nπ₃ sin_2n^π

√n .

(10) Determine whether the series is absolutely convergent, conditionally convergent or divergent. Explain your answer.

(a)

∞

X

n=1

(−1)ⁿ⁻¹

√4

n . (b)

∞

X

n=1

n 2 3

n

.

(c)

∞

X

n=1

1 + 1

n

ⁿ² .

(d)

∞

X

n=1

(−3)ⁿ (2n + 1)!. (e) X

n=1^∞

(2n)!

n!n!. (f)

∞

X

n=1

(−1)ⁿ n²+ 1 2n²+ 1

ⁿ