Mu Sequences and Series
National Convention 2012
For all questions, answer E. "NOTA" means none of the above answers is correct.
Note that for the following problems, i is defined such that i 2 = -1.
æ 1 öæ
1 öæ
1 ö æ
1 ö
1. Simplify the product ç1+ ÷ç1+ 2 ÷ç1+ 4 ÷ ç1+ 22012 ÷ .
è a øè a øè a ø è a ø
1-1 a 2
1-1 a
E) NOTA
2012
1-1 a 2
1-1 a
2013
A)
B)
1-1 a 2
1-1 a
2014
C)
1-1 a 2
1-1 a
2015
D)
2. A geometric series has terms A20 = e5 and A40 = e45 . Find A30 .
A) e10
B) e15
C) e 20
D) e 25
E) NOTA
-k
æ
F ö
3. Let Fn denote the n Fibonacci number. Evaluate the following sum: åç lim n+1 ÷
k=0 è n®¥ Fn ø
¥
th
A)
1+ 5
2
B)
2+ 5
2
C)
3+ 5
2
D)
4+ 5
2
E) NOTA
For questions 4-8 determine the convergence (divergence) of the given series.
n
¥
n+1 2 æ 4 ö
4. å(-1) n ç ÷
è 5ø
n=1
A) Converges Absolutely
B) Converges Conditionally
C) Diverges
D) Inconclusive
E) NOTA
¥
np
5. å(-1)
given 0 < p <1
n!
n=1
A) Converges Absolutely
B) Converges Conditionally
D) Inconclusive
E) NOTA
n
C) Diverges
¥
2n -1
) ( n + 3)
n=1 (
A) Converges Absolutely
D) Inconclusive
6.
ån n+2
¥
ln n
3n + 2
n=1
A) Converges Absolutely
D) Inconclusive
7.
å(-1)
¥
8.
B) Converges Conditionally
E) NOTA
C) Diverges
B) Converges Conditionally
E) NOTA
C) Diverges
B) Converges Conditionally
C) Diverges
n-1
å n ln n
n=2
1
A) Converges Absolutely
Mu Sequences and Series
D) Inconclusive
National Convention 2012
E) NOTA
2012
9. Let Pn =1+ 2 +
A)
1007
3
B)
åP
n
+ n . Evaluate
1009
3
C)
.
n=1
( 2012) ( 2013)
1012
3
D)
1015
3
E) NOTA
10. Given a differentiable function f ( x ) = e x , determine the values of x such that the series
dy æ d 2 y ö æ d 3 y ö
1+ + ç 2 ÷ + ç 3 ÷ +
dx è dx ø è dx ø
converges.
A) (-¥,¥)
C) ( 0,¥)
2
3
B) (-¥, 0)
D) (-e, e)
E) NOTA
æ k æ 2 ö 3ö
11. Define ak = k ç åç 2 ÷ - ÷ for all k ³ 3. Find lim an .
n®¥
è n=2 è n -1 ø 2 ø
A) -4
B) -3
C) -2
D) -1
E) NOTA
æ1 1
1
1
12. Evaluate cos ç +
+
+
+
è p 4p 9p 16p
3
1
1
A) B) C)
2
2
2
ö
÷.
ø
3
2
D)
E) NOTA
n
13. Determine the value of the following limit: lim å
A)
p
2
B)
p
3
C)
1
.
2
n®¥
k
k=0 n +
n
p
D)
4
1 1 1
14. Evaluate 1- + - +
2 4 8
1
1
2
A)
B)
C)
3
2
3
D)
p
5
3
2
E) NOTA
E) NOTA
n2 ( x - 8)
15. Determine the interval of convergence of å
.
n2 - 6
n=0
¥
A) ( 5, 7)
B) ( 7, 9)
C) ( 9,11)
16. Evaluate the sum 2 + ip -
D) (11,13)
p 2 ip 3 p 4
2!
-
3!
+
4!
+...
5n
E) NOTA
Mu Sequences and Series
A) 1
B) 3
C) 5
National Convention 2012
E) NOTA
D) 7
17. Find the sum of the coefficients of the terms up to degree 4 in the MacLaurin series of
cos x
.
1- x
A)
83
24
B)
85
24
C)
29
8
D)
18. Determine a Taylor Series about
n+1
-1) 2 2n-1 x 2n
(
A) å
+C
( 2n )!
n=1
n
¥
(-1) x 2n+1 + C
D) å
n=0 ( 2n + 2 )!
¥
89
24
E) NOTA
x = 0 for the following integral
n +1
-1) 2 2 x n
(
B) å
+C
( 2n + 1)!
n =1
sin ( x )
ò x dx .
-1) x 2n+1
(
C) å
+C
n=0 ( 2n +1) ( 2n +1)!
¥
¥
n
E) NOTA
¥
n 2 7n
.
n!
n=0
19. Find the value of å
A) 56e 7
B) 34e8
C) 64e9
D) 49e10
E) NOTA
20. Determine the common ratio of an infinite geometric series with first term i -1 and
sum 5i + 7 .
15 - 5i
42 - i
38 - 6i
22 - 4i
A)
B)
C)
D)
E) NOTA
33
18
37
7
21. Find a power series, centered at x = 0 , for f ( x ) =
¥
A)
åx
¥
3n+1
B)
n=0
åx
¥
2n+1
n=0
22. Given the sequence an =
A) -
1
2
C)
B) 0
åx
x
.
1- x 3
¥
n+1
D)
n=0
åx
n
E) NOTA
n=0
n n e nx
a
, for what value of x will n+1 converge to
n!
an
1
C)
D) 1
E) NOTA
2
1
23. Evaluate 1+
1
1+
2+ 5
B)
2
1+
A)
1+ 5
2
C)
3+ 5
2
D)
4+ 5
2
E) NOTA
e?
Mu Sequences and Series
National Convention 2012
24. The first 3 terms of the binomial expansion of 9 - x are
x x2
x x2
x x2
A) 3- +
B) 3- C) 4 - 2 216
6 216
6 64
E) NOTA
¥
25. Evaluate
å
(-1)
n
2n
n!
2
B) e -1
x x2
D) 4 + 2 216
.
n=1
A) e 2 +1
C) e-2 +1
D) e-2 -1
E) NOTA
26. If the sequence {an } is recursively defined by a1 = 2 and an+1 = an + 2n for n ³1then find
a100 .
A) 9900
B 9901
C) 9902
D) 9903
E) NOTA
n
æ 3ö
27. Let A = åç ÷ . Evaluate the error if the first one hundred terms are used to
è ø
n=0 8
approximate the summation.
¥
A)
397
5× 899
B)
398
5× 899
C)
¥
28. For what value of k will
A) k < 1
E) NOTA
399
5× 899
åk
B) -1 < k £ 1
n=0
-n
D)
3100
5× 899
E) NOTA
converge?
C) (-¥,-1) È (1,¥)
D) (-¥,¥)
æ pö
p
29. The coefficient of ç x - ÷ in the Taylor expansion of y = cos x about x = is
è
4ø
4
2
2
2
2
A)
B)
C)
D)
E) NOTA
12
24
36
48
4
30. Evaluate
¥
å
n=1
( 3)
A) ln 2
( -1)n+1
n3n
( 3)
B) ln 4
C) ln ( 2)
( 3)
D) ln 8
E) NOTA