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Some practice problems for test 2.
These are all taken from old exams or practice material.
There is some repetition in the problems, I just copied everything I could find!
A. Improper integrals, convergence of series, tests for convergence. (10.7-11.4)
- (a) Compute the value of the improper integral
2
3 / 2
(ln )
1
dx
x x
if it converges
(b) State whether
2
3 / 2
(ln )
1
k k
converges, and prove your result
(c) Draw a sketch of a curve and some rectangles to justify your answer to this question: which
is larger, the answer to (a), or to (b)? Being a little more clever, see if you can give both an
upper bound and a lower bound for the sum in (b), using your answer to (a).
- Let be the region between the curve
/ 2
2
x
y e
and the x-axis, x 0. Find the volume
obtained by revolving about the y-axis. (Method: use shells. ok, here's how to do it: the
volume of a thin cylindrical shell is 2x
/ 2
2
x
e
dx, so integrate this from 0 to to get the answer.
I suggest substitution u = x
2
- (a) Decide if
1
2
1
sin
dx
x x
x
converges or not without computing value, and explain.
(b) Compute the value of the improper integral
1
0
x ( 1 x )
dx
(the integrand is a
famous density in probability theory called the arcsine density). Method: First make the
change of variable x = u + 1/2 and change the limits of integration so that you have a
definite integral with the proper limits on u as you do this. Then make the change of
variable u = (1/2)sin , and change the limits of integration to be correct for (for
example, if u = 1/2, then must be /2). Then you can get the answer. This is better
than working your way all the way back to x in order to plug in the original limits.
2
(ln )
e
dx
x x
( make substitution u = ln x)
2
2
dx
x
(Hint: factor the denominator)
sin( )
3
1
x
e
dx
x
Use comparison test just to decide if converges;
Do not evaluate. (Hint: what is range of sin x ?)
7. Use integral test (you must do it this way) to prove convergence or
divergence:
2
2
(ln )
k
k k
8. (a) Use limit comparison to prove whether or not the series
1
2 1
3
k
k
converges ( pick
some series comparable to this for which you know whether or not it converges).
(b) Prove that the series
2
ln
( 1 )
k
k
k k
converges. Letting L be what it converges to, find an
upper bound for the absolute difference between L and the sum up through the k = 99 term; that
is, find an upper bound for | L - s 99
| , where s 99
99
2
ln
k
k
k k
- For each of these, explain why they do or don't converge:
(a)
ln( )
1
2
k k
(b)
ln( 1 )
1
1
k
k
(Hint: use limit comparison ).
(c)
k
k
k
2 k!
(d)
1
1
1
( 1 ) ln( 1 )
k
k
k
. If the first 100 terms of the series are summed, what is a
useful upper bound on the absolute difference between that sum and the sum of the
infinite series?
10. Find
1 1
0
k k
k
k
(Hint: write as sum of two series)
11. Use integral test (you must do it this way) to prove convergence or
divergence:
2
ln
k
k k
12. Use limit comparison test to prove convergence or divergence:
1/ 2
2
1
ln
k
k
k k
(Hint: compare to series 1/k
p
for some p).
13. Use root test to prove convergence or divergence:
1
k k
k
k
( see section 10.4, last box, to help with the limit)
14. Explain why this series converges or diverges
1
1
( 1) ln
k
k
k
k
- (a) compute
0
1
3
1 2
k k
k
(b) compute
2
( 1 )
1
k
k k
(b) Find the Taylor polynomial of degree 2 for the function f(x) =
x
1
, expanded about the
point a = 1 (that is, in powers of x-1 ). If x = 1.1, find an upper bound for the error term
| ( 1. 1 ) |
3
R
. (c) One way to get the Taylor series for
2
x
e
expanded about 0 is to just start taking
derivatives and evaluating at 0, etc. You will find that it is a lot of work if you do a few terms.
Can you think of a better way?
C. Power series, diff. and integration of power series, binomial expansion (11.7-11.9).
- Find the radius of convergence of the power series
k
k
x
k
k!
(Use the ratio test. Use your
algebra skill to arrange the ratio into something with a recognizable limit. Hint:
k
k
k
( k 1 )
can be
written as
k
k
k
1
and this can be written as a limit you learned in sec. 10.4).
- Find the radius of convergence of the power series
k
k
x
k k
2
4
( 1 )
- Find the radius of convergence of the power series
k
k
x
k
2
2
2
4. Find the radius of convergence of the power series k
1/
k
x
2k
5. Find the interval of convergence of the power series x
k
k
ln k )
(Find the radius of convergence, then test the endpoints of the interval separately).
- Find the radius of convergence of the power series
k
k
x
k
k
!
(ratio test)
- Let F(x) =
x
t dt
0
cos( )
(a) Expand F(x) in powers of x, showing terms up through the x
4
term.
(method: expand cos( t ) using the known expansion for cos but substituting t , showing
terms up through t
3
term, and integrate term by term)
(b) Using your series for F(x), find F
(4)
(0). (same trick as exercise 9 of homework )
(c) Approximate F(1) by using the series from (a) up through the x
3
term. Give an upper bound
on the absolute error of your approximation (note the series for F(1) is alternating with dec.
terms).
- Let F(x) =
x
dt
t
t
0
sin
(a) Expand F(x) in powers of x, showing terms up through the x
5
term.
(method: expand sin(t)/t by using the known expansion for sin, showing terms up through t
4
term, and integrate term by term)
(b) Using your series for F(x), find F
(5)
(0). (same trick as exercise 9 of homework )
(c) Approximate F(1) by using the series from (a) up through the x
3
term. Give an upper bound
on the absolute error of your approximation (note the series for F(1) is alternating with dec.
terms).
9. (a) Expand the function f(x) = (sin x)/ x in powers of x ; you can just write out
4 or 5 terms of the series to show the pattern, if you don’t like summation notation.
(Use the known expansion for sin x and divide by x to do this, rather than
differentiating f repeatedly !! This is the trick).
(b) Find f
(20)
(0). (Note the connection between f
(n)
(0) and the coefficient of x
n
in the series expansion of f : this is called the Taylor series for f; see formula
11.5.4, or see pg. 700, 2/3 of the way down the page. Note that you already have
the series from part (a), so you should be able to figure out what f
(n)
(0) is from
that!)
(c) The function S(x) =
0
sin
x
t
dt
t
appears in many engineering
problems. Use the expansion you developed in (a), integrating term by term,
to estimate S(1) with an error of no more than .00005.
( To estimate the error here, rather than trying to use Lagrange formula for
remainder, instead use the very helpful fact from section 11.4 that for a
convergent alternating series of terms of decreasing magnitude, the error made in
using finitely many terms of the series is no larger than the first neglected term
(11.4.5). Apply this to the integrated series. )
- (a) Write out terms up through the x
12
term for the series expansion in powers of x for the
function f(x) = cos (x
2
). (Hint: do not do this from scratch! Instead use the expansion you
know for cos, plugging in x
2
in place of x ). (b) Find f
(8)
(0). (Don’t do this by just
differentiating cos (x
2
) eight times - way too hard! Instead, recall how the coefficient of the x
8
term in the series expansion of f is related to f
(8)
(0); see page 700). This is a nice trick for
what would have been a difficult calculation if done directly. If you don't believe it, try to
differentiate cos(x
2
) eight times.
- Recall the series expansion (valid for –1 < x 1)
/ 2 / 3 / 4 ln( 1 )
2 3 4
x x x x x
From this,obtain a formula for the series
2 1 3 2 4 3 5 4
2 3 4 5
x x x x
(use the integral table in the book to integrate ln(1+x); to find the constant of integration,
note that the series is 0 when x=0, so you must choose C so that is true for the right side
also). The radius of convergence of this power series is 1, but your formula is valid at
x=1 (why? ..Abel's theorem on page 697). Using this, give the exact value for the sum
