Quiz 4 (Practice) Name
Math 253
July 21, 2009
Show all work. Answers without adequate justification will not receive credit.
For problems 1-7, assume
โ
X
n=1
an= 5.
Decide if the following statements are ALWAYS true, SOMETIMES true, or NEVER true. Circle
the appropriate answer.
1.
๎
๎
๎
๎
ALWAYS SOMETIMES NEVER The sequence {an}converges.
The sequence must converge to 0 since the series converges.
2.
๎
๎
๎
๎
ALWAYS SOMETIMES NEVER The sequence {an}is summable.
Saying a sequence is summable is the same as saying the infinite series converges.
3. ALWAYS
๎
๎
๎
๎
SOMETIMES NEVER The sequence {an}is absolutely summable.
โ
X
n=1
5(โ1)n+1
nln(2) =5
ln(2)
โ
X
n=1
(โ1)n+1
n= 5, but โ
X
n=1 ๎
๎
๎
๎
5(โ1)n+1
nln(2) ๎
๎
๎
๎
=โ
4. ALWAYS SOMETIMES
๎
๎
๎
๎
NEVER The sequence {an}diverges.
5. ALWAYS SOMETIMES
๎
๎
๎
๎
NEVER The sequence Sk=Pk
n=1 anis summable.
The limit of the sequence {Sk}is 5, so the series Pโ
k=1 Skcannot converge.
6. ALWAYS
๎
๎
๎
๎
SOMETIMES NEVER The sequence {(โ1)nan}is summable.
e.g. an= (โ1)n1
n.
7.
๎
๎
๎
๎
ALWAYS SOMETIMES NEVER The sequence {nโ2an}is summable.
Since anโ0 as nโ โ, the sequence anis bounded. In particular, there is some number M > 0
such that |an|< M for all n. Now we have
๎
๎
๎
an
n2๎
๎
๎=|an|
n2โคM
n2
Since {M
n2}is summable, the sequence {๎
๎nโ2an๎
๎}is summable by the comparison test. In other
words {nโ2an}is absolutely summable, and thus it is also summable.
