Calculus I

Choi Nari

week11 1
8. Series
8.4 Other Convergence Tests
❖ Alternating Series
➢ An alternative series is a series whose terms are alternatively positive and negative.

We see from these examples that the nth term of an alternating series is of the form

where 𝑏𝑛 is a positive number.

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8. Series
8.4 Other Convergence Tests
❖ Alternating Series

The following test says that if the terms of an alternating series decrease toward 0 in absolute value, then
the series converges.

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8. Series
8.4 Other Convergence Tests
❖ Alternating Series

➢ The idea of the proof:

We first plot 𝑠1 = 𝑏1 on a number line. To find 𝑠2 we subtract 𝑏2 , so 𝑠2 is to the left of 𝑠1 .

Then to find 𝑠3 we add 𝑏3 , so 𝑠3 is to the right of 𝑠2 . But, since 𝑏3 < 𝑏2 , 𝑠3 is to the left of 𝑠1 .
Continuing in this manner, we see that the partial sums oscillate back and forth.
Since 𝑏𝑛 → 0, the steps are becoming smaller and smaller.
The even partial sums 𝑠2 , 𝑠4 ,… are increasing and the odd partial sums 𝑠1 , 𝑠3 ,… are decreasing.
Thus it seems that both are converging to some number s, which is the sum of the series.

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8. Series
8.4 Other Convergence Tests
❖ Alternating Series
(−1)𝑛 3𝑛
Example 2. Test the alternating series σ∞
𝑛=1 for convergence or divergence.
4𝑛−1

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8. Series
8.4 Other Convergence Tests
❖ Alternating Series
(−1)𝑛+1 𝑛2
Example 3. Test the alternating series σ∞
𝑛=1 𝑛3 +1
for convergence or divergence.
Soln.

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8. Series
8.4 Other Convergence Tests
❖ Estimating Sums of Alternating Series

The following Theorem says that for series that satisfy the conditions of the Alternating series test, the size of
the error is smaller than 𝑏𝑛+1, which is the absolute value of the first neglected term.

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8. Series
8.4 Other Convergence Tests
❖ Estimating Sums of Alternating Series
Example 4.
Soln.

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8. Series
8.4 Other Convergence Tests
❖ Absolute Convergence

Given any series σ 𝑎𝑛 , we see that

σ∞𝑛=1 |𝑎𝑛 | = 𝑎1 + 𝑎2 + 𝑎3 + ⋯
whose terms are the absolute values of the terms of the original series.

➢ If σ 𝑎𝑛 is a series with positive terms, then | 𝑎𝑛 |= 𝑎𝑛 and so absolute convergence is the same as
convergence in this case.

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8. Series
8.4 Other Convergence Tests
❖ Absolute Convergence
Example 5.

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8. Series
8.4 Other Convergence Tests
❖ Absolute Convergence
(−1)𝑛−1 1 1 1
Example 6. The alternating harmonic series σ∞
𝑛=1 𝑛
=1−2+3−4+⋯

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8. Series
8.4 Other Convergence Tests
❖ Absolute Convergence

➢ The following theorem says that absolute convergence implies convergence.

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8. Series
8.4 Other Convergence Tests
❖ Absolute Convergence
Example 7. Determine whether the series is convergent, or divergent.

Soln.

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8. Series
8.4 Other Convergence Tests
❖ The Ratio Test

➢ The following test is very useful in determining whether a given series is absolutely convergent.

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11. Sequences, Series, and Power series
8.4 Other Convergence Tests
❖ The Ratio Test

Example 3. Part (iii) of the Ratio Test says that if then the test gives no information.
For instance, let’s apply the Ratio Test to each of the following series:

In both cases the Ratio Test fails to determine whether the series converges or diverges, so we must try another
test. Here the first series is the harmonic series, which we know diverges; the second series is a p-series with
p>1, so it converges.

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8. Series
8.4 Other Convergence Tests
❖ The Ratio Test
𝑛3
Example 8. Test the series σ∞ 𝑛
𝑛=1(−1) 3𝑛 for absolute convergence.
Soln.

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8. Series
8.4 Other Convergence Tests
❖ The Ratio Test
Example 9.
Soln.

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8. Series
8.4 Other Convergence Tests
❖ The Root Test
The following test is to apply when nth powers occurs.

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8. Series
8.4 Other Convergence Tests
❖ The Root Test
Example 10. Test the convergence of the series
Soln.

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H.W.6,10,12,14,28,35,38 8.4 19
8. Series
8.4 Other Convergence Tests
Determine whether the series are convergent of divergent.
3 4𝑛 𝑛
(a) σ∞
𝑛=1 1+ 𝑒 −2𝑛 (b) σ∞
𝑛=2
𝑛 𝑛+1 !

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8. Series
8.5 Power Series
❖ Power Series
➢ A power series is a series of the form
where 𝑥 is a variable and the 𝑐𝑛 ′s are constants called the coefficients of the series.

For each fixed 𝑥, the series (1) is a series of constants that we can test for convergence or divergence.
A power series may converge for some values of 𝑥 and diverge for other values of 𝑥.
The sum of the series is a function

whose domain is the set of all 𝑥 for which the series converges.

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8. Series
8.5 Power Series
❖ Power Series

➢ A series of the form

is called a power series in (𝑥 − 𝑎) or power series centered at 𝑎 or a power series about 𝑎.

➢ We have adopted the convention that (𝑥 − 𝑎)0 = 1 even when 𝑥 = a.

➢ When 𝑥 = 𝑎, all of the terms are 0 for 𝑛 ≥ 1 and so the power series always converges when 𝑥 = 𝑎.

➢ To determine the value of x for which a power series converges, we normally use the Ratio (or Root) Test.

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8. Series
8.5 Power Series
❖ Power Series
Example 1. For what values of 𝑥 is the series σ∞ 𝑛
𝑛=0 𝑛! 𝑥 convergent?
Soln.

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8. Series
8.5 Power Series
❖ Power Series
(𝑥−3)𝑛
Example 2. For what values of 𝑥 is the series σ∞
𝑛=0 convergent?
𝑛
Soln.

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8. Series
8.5 Power Series
❖ Power Series
(−1)𝑛 𝑥 2𝑛
Example 3. Find the domain of the Bessel function of order 0 defined by 𝐽0 𝑥 = σ∞
𝑛=0 2𝑛
2 2
(𝑛!)
Soln.

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8. Series
8.5 Power Series
❖ Interval of Convergence

➢ The number R in case (iii) is called the radius of convergence of the power series. R=0 in case (i) and R=∞ in case(ii).
➢ The interval of convergence of a power series is the interval that consists of all values of x for which the series
converges.
➢ In case (i) the interval consists of a single point a.
➢ In case (ii) the interval is (−∞, ∞).
➢ In case (iii) note that the inequality 𝑥 − 𝑎 < 𝑅 can be written as a-R<x<a+R.

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8. Series
8.5 Power Series
❖ Interval of Convergence

When 𝑥 is an endpoint of the interval, 𝑥 = 𝑎 ± 𝑅, anything can happen-the series might converges at one or
both endpoints or it might diverge at both endpoints. Thus, there are four possibilities for the interval of convergence:

➢ Note: In general, the Ratio Test (or the Root Test) should be used to determine the radius of convergence R.
The Ratio and Root Test always fail when 𝑥 is an endpoint of the interval of convergence, so the endpoints must
be checked with some other test.

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8. Series
8.5 Power Series
❖ Interval of Convergence
Example 4. Find the radius of convergence and interval of convergence of the series
Soln.

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8. Series
8.5 Power Series
❖ Interval of Convergence
Example 4. Find the radius of convergence and interval of convergence of the series
Soln.

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8. Series
8.5 Power Series
❖ Interval of Convergence
Example 5. Find the radius of convergence and interval of convergence of the series
Soln.

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8. Series
8.5 Power Series
❖ Interval of Convergence
Example 5. Find the radius of convergence and interval of convergence of the series
Soln.

H.W. 6,8,18,20,22,32,36
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