MTH 201 Other Sample
Lebanese American University · Calculus III · 2 pages · Dr. May Hamdan
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Sample Exam 3
Calc III
Summer I, 2013
1. Determine if the series below converge or diverge.
(a)
1
X
n=2
1
n (ln n)3
(b)
1
X
n=1
�
1 + 3
n
�n
(c)
1
X
n=1
(0:8)n
n!
(d)
1
X
n=1
jsin nj
n(n2 + pn + 1)
(e)
1
X
n=1
en
1 + e2n
(f)
1
X
n=1
n
(n1=n + 8)n
2. Use series to write the number 0:213 as a ratio m
n of two nonzero integers.
3. Find two series:
1
X
n=1
an and
1
X
n=1
bn with positive terms such that:
an ! 0; bn ! 0;
1
X
n=1
an converges,
1
X
n=1
bn diverges
4. Consider the series:
1
X
n=1
��1
2
�n
(a) Find the fourth partial sum s4.
(b) Estimate the magnitude of the error involved in using s4 to approximate the sum of the entire
series
(c) Find the sum of the series:
(d) Deduce using parts a and c above the exact error involved in using s4 to approximate the sum of
the entire series.
5. Find the sum of the series:
1
X
n=1
3
(n + 1)(n + 2)
6. Determine if the alternating series below converge absolutely, conditionally or diverge.
(a)
1
X
n=1
(�1)n(n + 5)
n2 + 1
(b)
1
X
n=1
(�1)n np
e�2
1
(c)
1
X
n=1
(�6)�n
7. For what values of x do the following series converge?
(a)
1
X
n=1
(x � 2)n
n
(b)
1
X
n=1
��
1 + 3
n
�n
xn
�
8. BONUS : Give an example of a power series that converges for all x 2 R:
2