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
{ a n
} is a sequence of non-negative terms. Show that its sequence of partial sums s n
= a
1
+ 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
≤
1 for each n
≥
1 and find the limit of this sequence.
a n 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.