MATH 121 Final: MATH 121 Amherst S12M121 2802 29ChingFinal
15 Feb 2019
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Department
Course
Professor
Math 121 Final Exam December 22, 2017
•This is a closed-book examination. No books, notes, calculators, cell phones, communication
devices of any sort, or other aids are permitted.
•Numerical answers such as sin π
6, 43
2,eln 4, ln(e7), e−ln 5,e3 ln 3, arctan(√3), or cosh(ln 3)
should be simplified.
•Please show all of your work and justify all of your answers. (You may use the backs of pages for
additional work space.)
1. [20 Points] Evaluate the following limits. Please justify your answer. Be clear if the limit
equals a value, +∞or −∞, or Does Not Exist. Simplify.
(a) lim
x→0
(sin(3x)) −3x
x−arctan x(b) Compute lim
x→0
(sin(3x)) −3x
x−arctan xagain using series.
(c) lim
x→∞ x
x+ 1x
2. [10 Points] Evaluate the following integral.Zcos x
4 + sin2x5
2
dx
3. [40 Points] For the following improper integral, determine whether it converges or diverges.
If it converges, find its value. Simplify.
(a) Z1
0
x3+ 4x+ 3
x3+ 3xdx =Z1
0
x3+ 4x+ 3
x(x2+ 3) dx (b) Z5
−∞
6
x2−4x+ 7 dx
(c) Z∞
6
6
x2−4x−5dx (d) Ze
0
ln x
√xdx
4. [18 Points] Find the sum of each of the following series (which do converge). Simplify.
(a)
∞
X
n=1
(−1)n5n+1
23n−1(b)
∞
X
n=0
(−1)n+1 2n+1 (ln 9)n
n!(c)
∞
X
n=0
(−1)nπ2n
9n−1(2n+ 1)!
(d) 1
3−1
5+1
7−1
9+. . . (e) 1−π2
(4)2! +π4
(16)4! −π6
(64)6! +π8
(256)8! −. . . (f) −1+ 1
2−1
3+1
4−1
5+. . .
5. [30 Points] In each case determine whether the given series is absolutely convergent,
conditionally convergent, or divergent. Justify your answers.
(a)
∞
X
n=1
(−1)n(√n+ 5)
n2+ 2 (b)
∞
X
n=1
(n+ 4)2
ln(n+ 4)
(c)
∞
X
n=1
(−1)nn+ 1
n2(d)
∞
X
n=1
(−1)n(3n)! ln n
(n!)2e4nnn
1
Document Summary
December 22, 2017: this is a closed-book examination. 3: [20 points] evaluate the following limits. Be clear if the limit equals a value, + or , or does not exist. 2 cos x (cid:0)4 + sin2 x(cid:1: [40 points] for the following improper integral, determine whether it converges or diverges. 6 x3 + 4x + 3 x3 + 3x. 6 x2 4x 5 dx = z 1. 0 x3 + 4x + 3 x(x2 + 3) dx (b) z 5 dx (d) z e. 6 x2 4x + 7 dx: [18 points] Find the sum of each of the following series (which do converge). In each case determine whether the given series is absolutely convergent, (a) (c) Xn=1 ( 1)n ( n + 5) n2 + 2 n2 (cid:19) ( 1)n(cid:18) n + 1 (n + 4)2 ln(n + 4) ( 1)n(3n)! ln n (n! 1: [15 points] find the interval and radius of convergence for the following power series.
