Math 8 Quiz; Due 07/23/14, 01:20 p.m. S. Soleymani
Instructions:
Write answers neatly on separate sheets of good-quality paper. Do not cramp results on
one or two pages for the sake of saving paper!
You must show your own work even if you work with others or if you take advantage of
online resources.
Sufficient explanation and proper step-wise reasoning is needed when you perform a test of
convergence or divergence of a series.
For Problems 1 and 2 use the following theorem: If { }na is convergent, then 1lim lim .n n
n n
a a
1. A sequence { }na is given recursively by 1 12, 2 .n na a a
(a) Write down the first four terms of the sequence.
(b) It is known that { }na is a convergent sequence. Find lim .n
n
a
2. A sequence { }na is given recursively by 1 1
1
1, 1 .
1n n
a a
a
(a) Write down the first four terms of the sequence.
(b) It is known that { }na is a convergent sequence. Find lim .n
n
a
(c) Use the results from (a) and (b) to estimate 2.
3. Find a formula for the nth partial sum nS of the series and use it to determine if the series converges
or diverges. If a series converges, find its sum by evaluating lim .n
n
S
(a)
1
4 3
n
n n
(b) 2
1
6
4 1
n
n
(c)
1
1 1
ln( 2) ln( 1)
n
n n
(d) 1
3
cos
3(5 )n
n
n
4. Consider the series
1
( 1)!
n
n
n
(a) Use the ratio test to show that the series converges.
(b) Find and express each partial sum 1 2 3 4, S , S , and SS as a reduced fraction. Use the pattern to
guess a formula for nS .
(c) Find the sum.
5. Determine if the following series converge absolutely, converge conditionally, or diverge. You
need to check the requirements for a test being used. For instance, if integral test is being used then
you need to first show that the sequence is positive and eventually a decreasing function of n.
Remember, there may be more than one way to determine the series’ convergence or divergence.
(a)
2
3
(1 )
(ln ) (ln ) 1n
n
n n
(b)
2
1
sin
2n
n
n
(c)
2
3 2
1
(ln )
n
n
n
(d)
3
( 1)
ln(ln )
n
n
n
(e)
2
1
2
2
n
n
n
n
n
(f)
1
1
sin n
n
n
(g)
2
2
1
( 2)!
!(3 )n
n
n n
n
(h) 2
1
(2 )
n
n
n
n
(i)
3
2
log ( !)n
n
n
n
(j)
1
( 1)
(arctan )
n
n
n
n
6. Show that lim 0.
(2 )!
n
n
n
n
7. Solve the equation 2 31 5x x x for x.
8. Use the following power series for
0
1
, | | 1
1
n
n
x x
x
to find a power series representation for
1
( ) ln
1
x
f x
x
near 0.c The power series must be in the
form
0
n
n
n
a x
...
1. Math 8 Quiz; Due 07/23/14, 01:20 p.m.
S. Soleymani
Instructions:
-quality
paper. Do not cramp results on
one or two pages for the sake of saving paper!
or if you take advantage of
online resources.
icient explanation and proper step-wise reasoning is
needed when you perform a test of
convergence or divergence of a series.
For Problems 1 and 2 use the following theorem: If { }na is
convergent, then 1lim lim .n n
n n
1. A
2. (a) Write down the first four terms of the sequence.
(b) It is known that { }na is a convergent sequence. Find lim
.n
n
a
2. A sequence { }na is given recursively by 1 1
1
1, 1 .
1n n
a a
(a) Write down the first four terms of the sequence.
(b) It is known that { }na is a convergent sequence. Find lim
.n
n
a
3. (c) Use the results from (a) and (b) to estimate 2.
3. Find a formula for the nth partial sum nS of the series and
use it to determine if the series converges
or diverges. If a series converges, find its sum by evaluating
lim .n
n
S
1
4 3
n
n n
1
6
4 1
n
n
5. 4. Consider the series
1
( 1)!
n
n
n
(a) Use the ratio test to show that the series converges.
(b) Find and express each partial sum 1 2 3 4, S , S , and SS
as a reduced fraction. Use the pattern to
guess a formula for nS .
(c) Find the sum.
5. Determine if the following series converge absolutely,
converge conditionally, or diverge. You
need to check the requirements for a test being used. For
instance, if integral test is being used then
you need to first show that the sequence is positive and
6. eventually a decreasing function of n.
Remember, there may be more than one way to determine the
series’ convergence or divergence.
(a)
2
3
(1 )
(ln ) (ln ) 1n
n
n n
2
1
sin
2n
n
n
