Math 42 — Chapter 8 Practice Problems
Set A
1.
(a) Write the series:

3
1
+
5
4

7
9
+
9
16

11
25
+
13
36

15
49
+
· · ·
in sigma notation.
(You don’t have to investigate convergence or divergence.)
(b) Determine whether the series
∞
n
=1
(

3)
n
+1
2
3
n
converges or diverges, and compute the
sum if it is convergent.
2. Determine whether the following series are convergent or divergent.
Indicate clearly which
tests you use and what conclusions you draw from them.
(a)
∞
n
=1
n
+ 5
5
n
(c)
∞
n
=1
3
n
n
2
n
!
(b)
∞
n
=1
(

1)
n
2
1
/n
(d)
∞
n
=2
1
n
√
ln
n
3. Find the interval of convergence of the power series
∞
n
=1
n
+ 1
n
3
(3
x
+ 2)
n
.
4. For this problem, we consider the series:
s
=
∞
n
=1
1
n
5
= 1 +
1
2
5
+
1
3
5
+
1
4
5
+
· · ·
(Interesting aside: one reason why we’d care about s is that it is the value of the socalled
Riemann Zeta Function
at 5: this function plays an important role in the field of number
theory, which concerns among other things the behavior of prime numbers, and surprisingly
has applications to things like secure Internet communication.)
(a) Explain why this is a convergent series; that is, explain why the number
s
is defined.
(b) If the first 10 terms of the series were used to approximate
s
, determine the accuracy of
this approximation. State your conclusion in a complete sentence, and be as quantita
tively precise as you can (but you do not need to simplify any expressions).
(c) It turns out that the sum of the first 10 terms of the series is the value 1.0369073413...
Use your reasoning from part (b) to obtain a more accurate approximation of
s
,
without
having to consider any more terms from the series.
Your answer does not need to be
simplified (or fully evaluated in decimal form).
5. Compute the Taylor series for ln(1 +
x
) about 0.
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6. Let
f
(
x
) =
√
x
.
(a) Find both the degree1 and degree2 Taylor polynomials for
f
about 100. (These func
tions are also called, respectively, the linear and quadratic approximations for
f
at 100.)
(b) Use the polynomials from part (a) to obtain two different approximations for
√
97.
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 '07
 Butscher,A
 Math, Calculus

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