Math 166 - Study Guide: Exam #4
• Positive Series: Similar to sequences, but the terms are summed together:
∞
X
ak .
k=1
1. Nth term test: If lim ak 6= 0, then the series diverges.
k→∞
2. Geometric Series: Consider
∞
X
ark .
k=1
– If |r| < 1, then the series converges and
∞
X
ark =
k=1
a
.
1−r
– If |r| ≥ 1, then the series diverges.
∞
X
1
.
3. P-Series: Consider
np
k=1
– Harmonic Series is a P-Series with p = 1, and it represents the cut-off for these types of
series.
– Harmonic Series diverges because the tail end does not go to zero fast enough.
– If p > 1 then the bottom grows faster than Harmonic Series, and we conclude the series
converges.
– If 0 < p ≤ 1 then the bottom grows slower than Harmonic Series, and we conclude the series
diverges.
4. Integral Test: The improper integral converges or diverges with the series. In order to apply
this test, you are first required to show:
– The terms are stricly positive on the interval.
– The function is continuous on the interval.
– The function is nonincreasing on the interval. Make sure you know how to spilt off terms of
the series if necessary to show this.
X
5. Ordinary Comparison Test: Compare the given series
ak to your over/under estimation
k
series
X
bk .
k
– Your estimate is generally a P-Series.
– You should state whether the series used for your over/under estimate converges or diverges.
– The main idea here is that a smaller divergent series implies the larger series diverges, or a
larger series converging implies the smaller series converges.
– Thus we
Xhave: X
X
X
∗ If
ak <
bk AND
bk convergent, then
ak convergent.
∗ If
k
X
k
bk <
k
X
ak AND
X
bk divergent, then
k
Xk
ak divergent.
k
6. Limit Comparison Test: Compares the given series
X
k
ak to your guess series
X
bk .
k
– Your estimate is generally a P-Series.
– You should state whether the series used for your guess converges or diverges.
– This test is useful if the given series involves a rational expression of polynomials and/or
radicals.
1
ak
represents how well you guess compares with the
bk
given series. (Ask yourself if you are comparing a Geo Metro with a Ferrari 430 Scuderia, or
are you comparing a Honda Accord with a Toyota Camry.)
∗ If 0 < L < ∞: (This is the Honda vs. the Toyota) Then your guess and the given series do
the same. Thus if your guess converges/diverges, then the given series converges/diverges
along with it.
∗ If L = 0 or L = ∞: (This is the Geo vs. the Ferrari) Then you picked a bad guess or did
something wrong, and you should find the mistake or chose a different test.
– There are actually two cases for L = 0, but you will not go wrong if you just assume this is a
bad thing. If you can differentiate the two cases, then please feel free to reach a conclussion
if you have the right case of L = 0.
– You are more likely to use this test over the ordinary comparison test since you DO NOT
need to worry about over/under estimating correctly.
– The main idea here is that L = lim
k→∞
7. Ratio Test: Compares the current term ak to the next term ak+1 of the given series.
ak+1
.
– Calculate: ρ = lim
k→∞ ak
– NOTE: The next term is ALWAYS over the current term.
– We then analyze ρ to determine the convergence or divergence of the series.
∗ If ρ < 1, then the next term is smaller than the current term. Thus, the series is
converging.
∗ If ρ > 1, then the next term is larger than the current term. Thus, the series is diverging.
∗ If ρ = 1, then the series is adding the same value over and over again, but we don’t
have any way telling if it is adding zero as required for convergence. Thus, this case is
inconclussive.
∗ Useful if the terms of the series include k!, rk , and/or k k
• Series with Negative Terms: The terms of the series can be both positive and negative.
1. Collapsing Sum: Look at the n
th
partial sum Sn =
n
X
ak .
k=1
– NOTE: Most of the terms should cancel each other out in the previous step.
∞
n
X
X
ak in order to evaluate the series.
ak =
– Then calculate the limit lim Sn = lim
n→∞
n→∞
k=1
k=1
2. Alternating Series Test: Determines the convergence/divergence of the Alternating Series
∞
X
k+1
(−1)
an .
k=1
– The Alternating Series converges if:
∗ The terms are decreasing. This is shown by proving either an+1 − an < 0 OR f 0 (k) < 0
where f (k) = ak .
∗ The terms go to zero. This is shown by proving lim ak = 0.
k→∞
– You may need to break off some of the terms from the series in order show this. Make sure
that you know how to do this, and why this does not alter the convergence/divergence of the
series.
– Types of Convergence:
∗ If the alternating series converges, then the series is at least conditionally convergent.
∞
X
∗ If
|x| converges, then the series is absolutely convergent.
k=1
2
– Make sure you know why the Alternating/Harmonic Series are important examples.
• Power Series:
– Know the following Series:
1
1.
= 1 + x + x2 + x3 + . . .
1−x
x3
x4
x5
x2
+
+
+
+ ...
2. ex = 1 + x +
2!
3!
4!
5!
3
5
7
x
x
x
3. sin (x) = x −
+
−
+ ...
3!
5!
7!
2
4
6
x
x
x
4. cos (x) = 1 −
+
−
+ ...
2!
4!
6!
– Alterations of Power Series to obtain different Power Series:
∗
∗
∗
∗
Add/Subtract polynomials/constant to a power series in order to obtain a new power series.
Mult/Divide a power series by polynomials/constants in order to obtain a new power series.
Mult/Divide a two power series in order to obtain a new power series.
Plug in (+/−)xc , for some constant c, into the x of a power series in order to obtain a new
powers.
∗ Integrate/Differentiata a power series to obtain a new power series. (NOTE: Alterations to
1
in order to obtain new power series expansions of rational
the power series expansion of
1−x
functions and logarithmic functions are especially important.)
• Other Series:
1. Taylor Series: Yields an approximation to the function f (x) about a value a.
∞
X
f (k) (a)
f 00 (a)
f 000 (a)
k
– f (x) =
(x − a) = f (a) + f 0 (a)(x − a) +
(x − a)2 +
(x − a)3 + . . .
k!
2!
3!
k=0
– You should also know how to find the nth order Taylor expansion for f (x) and estimate the
associated error term.
– Also, know how to solve for n in a general nth order Taylor expansion for f (x) in order to
find the value for n that yields a desired error bound.
2. Maclaurin Series: Just a Taylor Series with a = 0.
∞
X
f 00 (0) 2 f 000 (0) 3
f (k) (0) k
x = f (0) + f 0 (0)x +
x +
x + ...
– f (x) =
k!
2!
3!
k=0
– The rest is just identical to the Taylor Series.
• Convergence Sets and the Radius on Convergence:
1. Convergence Sets: The is the set of all values of x where the series converges.
– Generally, you find the convergence set using the absolute ratio test and then find the x values
that yield ρ < 1.
– Examples:
|x|
∗ If simplified absolute ratio test yields lim
= 0 then |x| is a constant and the limit is
k→∞ k
less than one ∀x ∈ (−∞, ∞).
3k |x|
∗ If simplified absolute ratio test yields lim
, then 3 |x| < 1 if −1 < 3x < 1 which
k→∞
k
yields the convergence set x ∈ (−1/3, 1/3).
3
∗ If simplified absolute ratio test yields lim 3k |x|, then lim 3k |x| < 1 if x = 0 which
k→∞
k→∞
yields the convergence set x = 0.
– Do NOT forget to check the endpoints of the interval individually to determine if they are in
the convergence set.
2. Radius of Convergence: It is always half the length of the interval on convergence.
• Other Notes:
– I will expect that you know the integrals/derivatives of the basic (polynomial, rational, logarithmic, exponential, roots, trig, and inverse trig) functions.
– You may want to know the identities
cosh(x) =
ex + e−x
2
sinh(x) =
ex − e−x
2
– Then,
d (cosh (x))
= sinh(x)
dx
d (sinh (x))
= cosh(x).
dx
– As always, it is very important that you READ THE DIRECTIONS for every question.
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