Berkeley City College
Math 3B - Calculus II - Chapters 11 - Sequences and Series
Name__________________________________________
Indicate whether the given series converges or diverges. If it converges, find its sum.
∞
8
1) ∑ (-1)k - 1
5k
k=1
1)
Write the given decimal as an infinite series, then find the sum of the series, and finally, use the result to write the
decimal as a ratio of two integers.
2) 0.9848484 . . .
2)
Solve the problem.
3) An object is rolling with a driving force that suddenly ceases. The object then rolls 10
meters in the first second, and in each subsequent interval of time it rolls 80% of the
distance it had rolled the second before. This slowing is due to friction. How far will the
object eventually roll?
4) A child on a swing sweeps out a distance of 16 ft on the first pass. If she is allowed to
continue swinging until she stops, and if on each pass she sweeps out a distance 3 of
4
3)
4)
the previous pass, how far does the child travel?
5) A ball is dropped from a height of 18 m and always rebounds 1 of the height of the
3
5)
previous drop. How far does it travel (up and down) before coming to rest?
6) Find the value of b for which 1 + eb + e2b + e3b + . . . = 2.
6)
Use the integral test to determine the convergence or divergence of the series.
∞
1
7) ∑
2k
k=1
7)
Use the limit comparison test to determine if the series converges or diverges.
∞ 6n2 + 4
8) ∑
n3 + 8
n=0
8)
Use the ratio test to determine if the series converges or diverges.
∞
9) ∑ n! e-8n
9)
n=1
10)
∞
∑
n=1
10(n!)2
(2n)!
Instructor: K Pernell
10)
1
Determine convergence or divergence of the series.
∞
n
8n
11) ∑
ln n + 5n + 6
n=1
12)
∞
∑
n=1
11)
1
12)
(5n + 1)3/2
Find the sum of the series as a function of x.
∞
13) ∑ (x - 2)n
n=0
14)
∞
x - 10 n
4
14)
x- 9 n
5
15)
(x - 4)2n
2n
n=0
16)
∑
n=0
15)
∞
∑
n=1
16)
13)
∞
∑
Find the Taylor series generated by f at x = a.
17) f(x) = x2 - 4x + 10, a = -1
17)
18) f(x) = 1 , a = 3
x2
18)
19) f(x) = ex, a = 7
19)
Find the power series representation for f(x).
20) f(x) = ln (1 + 4x)
20)
Solve the problem.
21) Use a Taylor series to estimate the integral's value to within an error of magnitude less
than 10-3.
∫
0.3
21)
cos2 x dx
0
22) Use a Taylor series to estimate the integral's value to within an error of magnitude less
than 10-3.
∫
0.2
ln(x2 + 1)dx
0
2
22)
Answer Key
Testname: MATH3B_CH11_PRACTICE
4
3
1) Converges;
2)
65
66
3) 50.0 m
4) 64 ft
5) 36 m
1
6) ln
2
7) Converges
8) Diverges
9) Diverges
10) Converges
11) Diverges
12) Converges
13) - 1
x- 3
14) -
4
x - 14
15) - x - 9
x - 14
16) -
2
(x - 4)2 - 2
17) (x + 1)2 - 6(x + 1) + 15
∞ (-1)n (n + 1)(x - 3)n
18) ∑
3n+2
n=0
∞ e7 (x - 7)n
19) ∑
n!
n=0
64 3
20) 4x - 8x2 +
x - 64x4
3
21) 0.2912
22) 0.002635
3