MATH 2414 - Exam 3 Review
Find the limit of the sequence or determine that the limit does not exist.
-9 n
1) a n = 1 +
n
2) a n =
n
6n · n
3) a n = n -
n 2 - 12n
Find the limit of the sequence or state that it diverges.
sin3 n
4)
3n
Find the sum of the series.
(-1)n-1
5)
n=1
6)
5
6n
1
1
n 7n
n=0 3
Determine if the series converges or diverges. If the series converges, find its sum.
1
n+1
7)
n=1
1
n+2
(tan-1 (n + 1) - tan-1 n)
8)
n=1
Use the integral test to determine whether the series converges.
9)
6n
2
n=1 n + 4
Use the ratio test to determine if the series converges or diverges.
10)
n=1
11)
n=1
8n
n!
10(n!)2
(2n)!
1
Use the root test to determine if the series converges or diverges.
n
10n 1/n - 1
12)
1/n - 1
n=1 3n
Use the Comparison Test to determine if the series converges or diverges.
1
n-1
+1
2
13)
n=1
Use the limit comparison test to determine if the series converges or diverges.
14)
9 n
3/2
- 7n + 7
n=1 3n
Determine convergence or divergence of the series.
n 4 e-n
15)
n=1
Determine convergence or divergence of the alternating series.
(-1)n
n 7/4
16)
n=1
Determine if the series converges absolutely, converges, or diverges.
17)
(-1)n
1/3 + 8
n=1 3n
18)
(-1)n
n=1
4n 2 + 1
4n 7 + 2
(-1)n ln
19)
n=1
9n + 7
8n + 6
Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire
series.
20)
n=1
(-1)n+1 (-0.2)2n+1
2n + 1
Find the series' radius of convergence.
21)
n=0
(x - 4)n
7n + 5
2
22)
n=1
(x - 7)n
(2n)!
Find the interval of convergence of the series.
23)
n=0
24)
n=0
(x - 9)n
5n + 4
(x - 9)2n
36n
Find the function represented by the power series.
25)
n=1
26)
n=0
x-1 n
2
x2 - 7
2
n
Find the Taylor series generated by f at x = a.
1
27) f(x) = , a = 3
x
Use power series operations to find the Taylor series at x = 0 for the given function.
28) f(x) = x5 sin x
29) f(x) = x10 tan-1 (5x)
Find the first four terms of the binomial series for the given function.
30) (1 + 9x)-1/2
3
Answer Key
Testname: EXAM 3 REVIEW
e-9
6
6
0
5
5)
7
1)
2)
3)
4)
6)
1
3
7) converges;
8) converges;
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
1
2
2
diverges
Converges
Converges
Diverges
converges
Diverges
Converges
Converges
Converges conditionally
Converges absolutely
Diverges
-1.86 × 10-9
1
, for all x
8 x < 10
3 < x < 15
x-1
25) x-3
26) -
2
2
x -9
27)
n=0
28)
n=0
29)
(-1)n (x - 3)n
3 n+1
(-1)n x2n+6
(2n + 1)!
(-1)n 5 2n+1 x2n+11
2n + 1
n=0
9
243 2 3645 3
x x
30) 1 - x +
2
8
16
4