Midterm 3: Sample Test
Math 1220-004
Fall 2014
Guidance:
(1) Chapter 9: 9.1 — 9.8
(2) The following tests are going to be provided during midterm 3 exam:
(a) Bounded Sum test (page 464)
(b) Integral test (page 464)
(c) Ordinary Comparison test (page 469)
(d) Limit Comparison test (page 470)
(e) Ratio test (page 471)
(f) Alternating Series test (page 475)
(g) Absolute Convergence test (page 476)
(h) Absolute Ratio Test (page 477)
(3) Things that do not appear on the above list should be memorized by you, including
but not limited to : definitions of the newly studied concepts in chapter 9, p-series
(page 465), geometric series (page 456), n-th term test for divergence (page 457),
how to find convergence set for a power series (section 9.5), Taylor and Maclaurin
series and formula, etc.
Sample Problems
1 Respond with true or false to each of the following assertions. If false, state why.
(1) An infinite series is the sum of all the terms in an infinite sequence.
∞
X
(2) For an infinite series
an , it could the case that lim an 6= 0.
n→∞
n=0
(3) If lim an = 0, then lim a2n = 0.
n→∞
n→∞
(4) If {an } and {bn } both diverge, then {an + bn } diverges.
∞
∞
X
X
(5) If
an converges, so does
a2n .
n=0
(6) If p ≥ 1, then the p-series
n=0
∞
X
n=0
1
converges.
np
(7) If bn ≤ an ≤ 0, for all natural numbers n, and
converges.
∞
X
n=0
bn converges, then
∞
X
n=0
an
(8) If
(9) If
∞
X
n=0
∞
X
an diverges, then
∞
X
|an | diverges.
n=0
an xn converges at x = −2, then it also converges at x = 2.
n=0
(10) Every power series converges for at least two values of the variable.
2 Calculate the following sums:
(1)
∞
X
k=0
(2)
1
1
[2( )k + 3(− )k ]
4
5
∞
X
1
1
( −
)
k k−1
k=2
3 Determine convergence or divergence for each of the positive series below. Indicate
the test you use and use it to prove your conclusions rigorously.
(1)
∞
X
n=1
(2)
n2
n
+ 3n + 1
∞
X
n!
n100
n=1
(3)
∞
X
3k + k
k=1
(4)
∞
X
k!
1
2 + sin2 n
n=1
4 For the following two problems, state whether the given series is absolutely convergent, conditionally convergent, or divergent.
(1)
∞
X
(−1)n
n=1
(2)
∞
X
1
3n − 1
(−1)n
n=1
3n
2n+8
5 Determine the convergence set for the following power series.
(1)
∞
X
(−2)n xn
n=0
(2)
∞
X
3n x3n
n=0
(3)
2n + 3
(3n)!
∞
X
n!(x + 1)n
n=0
3n
6 Find the first three nonzero terms of the Taylor series for each of the following. Also
indicate the set of convergence:
(1)
f (x) =
x
at a = 0
1 − 2x3
(2)
f (x) =
1
at a = 2
1−x
(3)
f (x) = x sin x at a = 0