# Analysis Homework #7 1. 2. 3. Analysis Homework #7

due Thursday, Mar. 10

1.

Define a sequence

{ a n

}

by letting

a

1

= 2 and

a n

+1

= 4

1

a n

for each

n

1.

Show that 2

≤ a n

≤ a n

+1

4 for each

n

1 and find the limit of this sequence.

2.

Suppose sums

s n

{ a n

}

=

a

1 is a sequence of non-negative terms. Show that its sequence of partial

+

a

2

+

. . .

+

a n

converges if and only if

s n

is bounded for all

n

.

3.

Define a sequence

{ a n

}

by letting

a

1

= 1 and

a n

+1

=

na

2

n n

+ 1 for each

n

1.

4.

Show that 0

≤ a n

+1

Suppose the series

∞ a n n

=1

≤ a n

1 for each

n

1 and find the limit of this sequence.

converges. Show that the series

∞ n

=1

1

1+

a n

diverges.

You are going to work on these problems during your Thursday tutorial (at 4 pm).

When writing up solutions, write legibly and coherently. Use words, not just symbols.

Write your name and then MATHS/TP/TSM on the first page of your homework.

Your solutions may use any of the results stated in class (but nothing else).

NO LATE HOMEWORK WILL BE ACCEPTED.