Calculous 2 Sequences & series Quiz
choid91
sequence_series_quiz.pdf
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
. Determine the interval and radius of convergence. Hint: ln ln ln .A A BB
9. Find the radius and interval of convergence of each power series.
(a) 0
( 3)
1
n n
n
x
n
(b) 1
( 2)n n
n
x n
