Stony Brook University
Mathematics Department
Julia Viro
Calculus II
MAT 132
Fall 2016
Practice Final
Final exam will cover sequences, series and power series (Sections 8.1-8.8) and will contain combined
problems, where you have to use all your calculus knowledge (differentiation, integration, and working
with power series).
A list of Maclaurin expansions of standard functions will be given on the Final exam (see the last
page).
Problems below are similar to the problems you may encounter on Final exam. For your convenience,
the problems are rubricated. Make sure that you can handle problems from all rubrics.
Comparison of asymptotic behavior of logarithm, power, exponential, and factorial
1. Find the following limits
ln n
a) lim √
n→∞ 3 n
b) lim
n→∞
n2016
2016n
c) lim
n→∞
πn
n!
d) lim
n→∞
ne
n!
Limit of a sequence
2. Determine whether the sequence {an }∞
n=1 converges or diverges. If itconverges,
n find the limit.3n
√
n4
2
n
−
3
n
a) an =
e)
a
=
b)
a
=
cos
πn
c)
a
=
n
d)
a
=
n
n
n
n
−3n5 + 2
n
32n+1
n 2
√
(−1) n
1
sin 2n
n
√
g) an =
h) an = n arctan
i) an = 32−3n
f ) an =
3
n
2
(−1) + 2n
n
2+ n
Recognizing the model series: geometric, telescoping, p-series
3. Determine whether the series converges or diverges. If it converges, find the sum. If it diverges,
explain why.
a)
∞ 22n + (−π)n
P
5n−1
n=1
b)
∞
P
1
2
n=1 n + 2n
c)
∞
P
n=1
n−1/3
d)
∞
P
n=1
ln
n
n+1
e)
∞
P
(ln 2)2n
n=1
Convergence tests: comparison, limit comparison, integral, ratio, root. Divergence test
4. For each of the series below, decide if it converges or diverges. You do not need to find the value
of the sum. To receive full credit, you must justify your answer.
∞
∞
∞
∞
∞
P
P
P
P
P
1
1
1
1
2n
√
b)
c)
d)
tan
e)
f
)
cos n2
2+3
n − 2n
n + 5n
n
3
n
ln
n
n
3
n=1
n=1
n=2
n=1
n=1
n=1
∞
∞
∞ (n!)2
∞
∞
∞ sin(n!)
P
P
P
P
P
P
1
1
1
1
4
3
−n
√
g)
h)
i)
j)
ne
k)
−
l)
n
2n
n! 3n
n3
0.1
n=1 n
n=1
n=1 (2n)!
n=1
n=1
n=1
a)
∞
P
2
Alternating series. Absolute and conditional convergence
5. Determine whether the series converge absolutely, converge conditionally, or diverge:
a)
∞ (−1)n+1
P
√
n
n=1
b)
∞ (−1)n n
P
n=1 2n + 1
c)
∞ (−1)n+1
P
nπ n
n=1
Radius and interval of convergence of power series
6. Find the radius and the interval of convergence of the following series:
a)
∞ (−2x + 3)n
P
√
3
n
n=1
b)
∞ (3x − 1)n
P
22n
n=0
c)
∞
P
(−1)n n! (x + 1)n
∞ (1 + en )
P
xn
n!
n=1
d)
n=0
Presentation of functions as power series
7. Present the following functions as power series and determine their intervals of convergence:
1
3x + 1
(in powers of x) b) 2
(in powers of x)
3x + 2
x + 2x − 3
1
d)
(in powers of x + 1)
2x − 3
a)
c)
x2
x
(in powers of x)
+9
8. For the given functions, find their power series (in powers of x) and the interval of convergence.
a) f (x) = ln(2x + 3)
b) f (x) =
1
(x + 2)3
c) f (x) =
x4
x
+ 2x2 + 1
Recognizing Maclaurin series of some important functions
9. Find the sums of the following series
a)
∞ 32n
P
n
n=1 2 n!
b)
∞ (−1)n π 2n
P
2n
n=0 6 (2n)!
c)
1
1
1
1
−
+
−
+...
3
5
1·2 3·2
5·2
7 · 27
d)
∞
P
1
n
n=1 n 2
e)
∞ n
P
n
n=1 3
Taylor and Maclaurin series and polynomials
10. Find the Taylor series for f (x) = x5 − 5x3 + x around x = 2.
11. Find the first two non-zero terms in Maclaurin series for f (x) = ln cos x. Draw the graph of the
function and its Maclaurin polynomial on the same coordinate system (you may use a grapher).
π
. Draw the graph the
2
function and its Taylor polynomial on the same coordinate system (you may use a grapher).
12. Find the Taylor polynomial of degree 2 for f (x) = sin x around x =
3
√
√
13. Find the Taylor polynomial of degree 2 for f (x) = 3 x around x = 8. Use it to approximate 3 9.
Draw the graph the function and its Taylor polynomial on the same coordinate system (you may
use a grapher).
14. Find the Maclaurin series of the following functions:
2
x
arctan
1 − cos x
4
2
a) e−3x
b)
c)
d) cos2 x
x
x2
Applications of power series to various problems in calculus
15. Explain why the formula
√
1+x≈1+
x x2
−
2
8
is valid for all 0 ≤ x ≤ 1.
3
sin x − x + x6
16. Evaluate the limit lim
x→0
x5
Z
x3
dx as a power series.
17. Evaluate the integral
1 − x7
Zx
18. Find power series expansion for the function f (x) =
et − 1
dt.
t
0
Z1/2
19. Evaluate the integral
x cos(x3 ) dx. Give your answer as a numerical series.
0
20. Let f (x) = x6 ex . Find f (2016) (0).
21. Let f (x) =
∞
P
n3n−1 xn−1 . Determine the domain of f . Draw the graph of f . Find the values of
n=1
f (1/6), f 0 (1/6) and
Z1/6
f (x) dx.
0
22. Let R be the region in the first quadrant which is bounded the graph of the function y =
and the lines y = 0, x = 0, and x = 1. Find the area of R as a numerical series.
23. Use binomial series to approximate the length of the arc of the curve y =
x = 1/2.
1 − cos x2
x4
1 3
x from x = 0 to
3
2
24. A plane region R is bounded by the lines x = 0, x = 2, y = 0 and the curve y = e−x . Find
the volume of the solid of revolution obtained when R is rotated around the x-axis. Express your
answer as a numerical series.
4
sin(x2 )
25. The region bounded by the curves y =
, y = 0, and x = 1 is rotated around the y-axis.
x
Find the volume of the solid of revolution as a numerical series.
26. Find first four non-zero terms of the power series expansion of the solution of the following initial
value problem

00
2

y − (1 + x )y = 0
y(0) = −2

y 0 (0) = 2
27. How to lose weight fast? Here is a calculus approach :o)
The weight w of a body of mass m depends on the hight h above the surface of the earth and is
given by the formula
mgR2
w(h) =
,
(R + h)2
where g is acceleration due to gravity and R is the radius of the earth. Use power series expansion
of w (in powers of h/R) to show that
2h
w(h) ≈ mg 1 −
.
R
Use this formula to estimate the height h to which you must bring your body in order to reduce
your weight by 0.1%. (Radius of the earth is about 6400 km.)
5
Answer Key (typos may occur)
1. a) 0
b) 0
2. a) 0
b) div
c) 0
5(20 + 3π)
5+π
3. a)
4. a) conv
d) 0
d) e−3 e) 0
c) 1
b)
3
4
c) div
f) 0
d) div
g) div
e)
h) 1
c) div
d) div
e) conv
f ) div
h) div
i) conv
j) conv
k) div
f ) conv
5. a) converges conditionally
b) diverges
1
27
(ln 2)2
1 − (ln 2)2
b) conv
g) conv
i)
c) converges absolutely
1
4
, (1, 2] b) , (−1, 5/3) c) converges at x = −1 only d) converges for all x
2
3
n
n
∞
∞
P
P (−1) 3 n
2(−1)n
x , |x| < 2/3
b)
− 1 xn , |x| < 1
7. a)
n+1
n+1
2
3
n=0
n=0
6. 1 a)
c)
∞ (−1)n
P
x2n+1 , |x| < 3
2n+2
3
n=0
8. a) ln 3 +
c)
∞
P
d) −
∞
P
2n
(x + 1)n , −7/2 < x < 3/2
n+1
5
n=0
∞ (−1)n+1 2n
P
xn , −3/2 < x ≤ 3/2
n
n3
n=1
b)
∞ (−1)n n(n − 1)
P
xn , |x| < 2
n+2
2
n=2
(−1)n+1 nx2n−1 , |x| < 1
n=1
√
3/2
1
d) ln 2
3) 3/4
2
10. −6 + 21(x − 2) + 50(x − 2)2 + 35(x − 2)3 + 10(x − 2)4 + (x − 2)5
9. a) e9/2 − 1
b)
c) arctan
x2 x4
−
2
12
1
π 2
12. 1 −
x−
2
2
1
1
13. 2 + (x − 8) −
(x − 8)2 , 2.08
12
288
∞ (−1)n 3n
∞
P
P
(−1)n
14. a)
x2n
b)
x4n+1
4n+2
n!
(2n
+
1)2
n=0
n=0
11. −
∞ (−1)n 22n−1
1 P
d) +
x2n
2 n=0
(2n)!
16. 1/120
17.
∞ x7n+4
P
+C
n=0 7n + 4
c)
(−1)n 2n
x
n=0 (2n + 2)!
∞
P
6
18.
∞
P
xn
n=1 n · n!
19.
(−1)n
6n+3
n=0 (2n)!(3n + 1)2
∞
P
2016!
2010!
1
1 1
, 4, 48,
21. − ,
3 3
3
20.
22.
(−1)n+1
n=1 (2n)!(4n − 3)
∞
P
23. 0.503
∞ (−1)n 23n+1
P
24. π
n=0 n!(2n + 1)
(−1)n
25. 2π
n=0 (2n + 1)!(4n + 3)
∞
P
26. y = −2 + 2x − x2 +
27. 3.2 km
x3
+ ...
3
Important Maclaurin Series
∞
P
1
=
xn = 1 + x + x2 + x3 + . . .
1 − x n=0
ex =
∞ xn
P
x
x2 x3
=1+ +
+
+ ...
1!
2!
3!
n=0 n!
ln(1 + x) =
(−1 < x < 1)
(for all x)
∞ (−1)n+1
P
x2 x3 x4
xn = x −
+
−
+ ...
n
2
3
4
n=1
( −1 < x ≤ 1)
x3 x5 x7
(−1)n
2n+1
sin x =
x
=x−
+
−
+ ...
3!
5!
7!
n=0 (2n + 1)!
∞
P
cos x =
∞ (−1)n
P
x2 x4 x6
x2n = 1 −
+
−
+ ...
2!
4!
6!
n=0 (2n)!
arctan x =
(for all x)
(for all x)
∞ (−1)n
P
x3 x5 x7
x2n+1 = x −
+
−
+ ...
3
5
7
n=0 2n + 1
∞ k(k − 1)(k − 2) . . . (k − n + 1)
P
xn =
n!
n=1
k(k − 1) 2 k(k − 1)(k − 2) 3
1 + kx +
x +
x + ...
2!
3!
( −1 ≤ x ≤ 1)
(1 + x)k = 1 +
(−1 < x < 1)