Math 126-104 Test 2 Sumer 2014 Carter
General Instructions: Write your name on only the outside of your blue book. Do all of your work and write your answers inside your blue book. Put your test paper inside your blue book as you leave. Solve each of the following problems. There are 115 points. Make sure your solution is written neatly, and I can recognize the answer. Fresh fruit in plain yogurt with just a little sugar can be pleasing.
1. Determine the limit of the sequence (5 points each) :
(a) a n
= ln n n
(b) a n
= n + 3 n 2 + 7 n + 6
(c) a n
= 2 + ( − 0 .
1) n
(d) a n
= n
2 n
2.
(10 points) Give a formula for the n th partial sum of the series
∞
X k =1 and use this formula to sum the series.
2 k 2
−
2
( k + 1) 2
,
3.
(5 points) Represent the repeating decimal
0 .
142857 = 0 .
142857142857142857 . . .
as a fraction in lowest terms.
4. Determine the interval of convergence for the power series (10 points each) .
(a)
∞
X
( − 1) n x 2 n +1
2 n + 1 n =0
(b)
∞
X n x n − 1 n =1

5. Use any test that you like to determine if the given series converges (8 points each) .
(a)
∞
X
1 n =2 n (ln ( n )) 2
(b)
∞
X
( n + 1) n !
n =1
(c)
∞
X
1 n =1 p n ( n + 1)( n + 2)
(d)
∞
X
1 n
√ n + 5 n =2
(e)
∞
X
( n !) 2 n =1
(2 n )!
6.
(10 points) Given that the geometric series
1 + y + y
2
+ · · · =
∞
X y n
=
1
1 − y
, n =0 use substitution ( y = − x , y = z
2
) and term-by-term integration to obtain a series for f ( x ) = ln ( x + 1) .
7.
(10 points) Use the series for f ( x ) = sin ( x ) to find a series for g ( x ) = x sin ( x
2
).