advertisement

**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.