Math 5318
Exam II
November 6, 2000
In Class Portion:
All outside references must be to definitions and/or theorems in Chapters 1.1 - 3.6 & Appendix.
4
1. Prove: Theorem 3.1B (part II): If
ja
n'1
n
converges to A and if c 0 ß, then
4
jca
n'1
n
converges
to cA.
sin
2. Find: a) lim
n64
n 4%1
n 2%1
n
b) lim
n64
2 n 2%1
2n&1
4
3. Suppose {s n}n'1 is a bounded sequence and lim inf sn ' m . Prove there exists a subsequence
4
{sn }k'1
k
n64
such that lim sn
k64
k
' m.
4. Give an example which shows that the conclusion of the Nested Interval Theorem may fail if
the hypothesis (in the theorem) that the intervals are nested is dropped.
4
5. Show that if the sequence {s n}n'1 is monotone non-decreasing, then for
the sequence {
4
}
n n'1
n
'
s1 % s2 %ã% s n
is monotone non-decreasing.
6. Determine whether the following series converge:
4
a)
j
k'1
k%1
2k%3
4
k
b)
j
2 % (&1)k
3
k'1
7. Find all real values x for which the series
4
(&1)k x k
k'1
2
j
k
%2
4
k
c)
converges.
j
k'2
2k % 3
k 3 & 2k % 1
n
Take Home Portion (Due Friday, Noon):
All outside references must be to definitions and/or theorems in Chapters 1.1 - 3.6 & Appendix.
1. Find lim xn
n64
x1 $ 0
a) where
0 # x1#1
b) where
xn%1 ' xn % 2 , n > 0
4
xn%1 ' 1 & 1 & xn , n > 0
4
2. Give examples of sequences {s n}n'1 and {tn}n'1 such that s n 6 0 and tn 6 4 and
a) s n tn 6 0
b) s n tn 6 3
c) s n tn 6 &3
d) s n tn 6 4
3. Determine whether the following series converge:
4
a)
j
k'1
4
c)
j
k'1
cos
4
1
b)
k2
k
j sin k1
2
k'1
k2
k%1
4
4. Find all real values of p such that the series
j
k'2
1
4
5. Find all real values of x for which the series
converges.
k log p k
j (3 % (&1) ) (x & 1)
k
k'1
k
converges.
6. Determine whether the following series converge absolute, converge conditionally or diverge:
4
a)
j
k'1
4
(&1) (&3) ã (1&2k)
1 @ 4 ã (3k&2)
b)
4
7. Estimate the value of the series
your estimate.
j
k'1
log 1%
j
k'1
(&1)k%1 k
k%1
(&1)k%1
to within an accuracy of 0.001. Justify
k3