Section(11.5( Suggested(reading(HW((not(collected)(( MATH ... · Section(11.5(...
6
Section 11.5 Suggested reading HW (not collected) MATH 1152Q Alternating Series Notes Page 1 of 6 TASK 1. Recall Strategy for series with positive terms Fill in the blanks by looking up the answers from the given pages. Determine whether the infinite series with positive terms converge or diverge. 1. (11.2 page 710) The Geometric Series → when has the form If , ____________________. If , ____________________. 2. (11.2 page 713) The Divergence Test If , ____________________. If , ____________________. 3. (11.2) Harmonic Series, special case of p-Series → = with p =1. 4. (11.2 Ex. 8) The Telescoping Series → when can be reduced to ______________. If exists, ______________. Otherwise, ______________. 5. (11.4 page 727) The Comparison Test → when none of the above methods works or when there is an obvious comparison. If and converges, _______. If and diverges, _______. If and converges, _______. If and diverges, _______. 6. (11.4 page 729) The Limit Comparison Test (most versatile) → when involves dominant terms (p-series, rational function, or a geometric series). Consider . If and converges, ________. If and diverges, ________. If and converges, ___________. If and diverges, ___________. If and converges, __________. If and diverges, __________. 7. (11.4 page 728) The p-Series → when has the form If , ____________________. If , ____________________. 8. (11.6) The Ratio Test → when involves factorials or powers. Consider . If , ______________. If , ______________. If , ______________. 1 k k a = 1 k k a = 1 k k r = 1 r 1 r < lim 0 k k a = lim 0 k k a 1 k k a = 1 1 p k k = 1 k k a = ( ) 1 1 k k k b b + = n S = lim n n S 0 k k a b < 1 k k b = 0 k k a b < 1 k k b = 0 k k b a < 1 k k b = 0 k k b a < 1 k k b = k a lim k k k a L b = 0 L < < 1 k k b = 0 L < < 1 k k b = 0 L = 1 k k b = 0 L = 1 k k b = L = 1 k k b = L = 1 k k b = 1 k k a = 1 1 p k k = 1 p 1 p > k a 1 lim k k k a r a + = 0 1 r < 1 r > 1 r =
Transcript of Section(11.5( Suggested(reading(HW((not(collected)(( MATH ... · Section(11.5(...
Section 11.5 Suggested reading HW (not collected) MATH 1152Q Alternating Series Notes
Page 1 of 6
TASK 1. Recall Strategy for series with positive terms Fill in the blanks by looking up the answers from the given pages.
Determine whether the infinite series with positive terms converge or diverge.
1. (11.2 page 710) The Geometric Series → when has the form
If , ____________________. If , ____________________.
2. (11.2 page 713) The Divergence Test If , ____________________. If , ____________________.
3. (11.2) Harmonic Series, special case of p-Series → = with p =1.
4. (11.2 Ex. 8) The Telescoping Series → when can be reduced to
______________. If exists, ______________. Otherwise, ______________.
5. (11.4 page 727) The Comparison Test → when none of the above methods works or
when there is an obvious comparison.
If and converges, _______. If and diverges, _______.
If and converges, _______. If and diverges, _______.
6. (11.4 page 729) The Limit Comparison Test (most versatile) → when involves
dominant terms (p-series, rational function, or a geometric series). Consider .
If and converges, ________. If and diverges, ________.
If and converges, ___________. If and diverges, ___________.
If and converges, __________. If and diverges, __________.
7. (11.4 page 728) The p-Series → when has the form
If , ____________________. If , ____________________.
8. (11.6) The Ratio Test → when involves factorials or powers. Consider .
If , ______________. If , ______________. If , ______________.
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Section 11.5 Suggested reading HW (not collected) MATH 1152Q Alternating Series Notes
Page 2 of 6
Alternating Series What if we are given a series where ? How do we determine whether the series converges? TASK 2. Go to Sec 11.5 page 732-733. Read until just before the “proof of the alternating series test” (reading the proof is optional). Alternating Series Test Definition Alternating Series Suppose that for all positive integer , then the series
is called the Alternating Series.
Example (Example 1 pg 734): is called the Alternating Harmonic Series.
TASK 3a. Complete the following by copying from the reading. Theorem The Alternating Series Test The alternating series converges provided
i.) is ____________________ for all . In other words, ______________________________.
ii.) ____________________.
TASK 3b. Procedure (fill in the blanks) If and ,
then ___________________________________________________________ (by the _____________________________________________ Test from Sec 11.2 page 713). If , check whether is decreasing.
• If is decreasing, then ________________________________________
(by the ___________________________________________ Test, Sec 11.5 page 732).
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+-å 0nb >
0nb > n
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11 n
nn
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+
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1
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Section 11.5 Suggested reading HW (not collected) MATH 1152Q Alternating Series Notes
Page 3 of 6
TASK 4 (Example 1 page 734): Answer the following question by closely following the explanation for Example 1 page 734 in the book. It would be more effective if you first write your answer without looking at the book then check the book afterwards.
Determine whether converges.
TASK 5. Complete the following. Theorem Alternating Harmonic Series
• The harmonic series _____________________(see Sec 11.2 page 713 Example 9).
• The alternating harmonic series _______________(see your answer above).
( ) 1
1
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1
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( ) 1
1
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-å
Section 11.5 Suggested reading HW (not collected) MATH 1152Q Alternating Series Notes
Page 4 of 6
TASK 6. Stay on page 734. Attempt to do Example 2 AND Example 3 on a separate paper without looking at the solution in the book. Check your solution with the book.
Please choose either Example 2 or Example 3 and write down its question and solution in the space below.
Section 11.5 Suggested reading HW (not collected) MATH 1152Q Alternating Series Notes
Page 5 of 6
Do TASKS 7 and 8 by imitating Example 1, 2, and 3 (pg 734). Hint: one of them converges and the other diverges. Check your answers with WolframAlpha.
TASK 7. Example:
Determine whether converges.
Step (ii): Calculate
Step (i): Estimate (in your head or scratch paper) whether is decreasing. If you believe it is decreasing, verify by:
• citing a known fact from class (for example, behavior of an exponential function - Sec 11.1 Example 11 pg 700 or Sec 11.1 Eq. 4 pg 697),
• computing a derivative (follow Sec 11.5 Example 3 page 734), • or directly (follow Sec 11.1 Example 13 page 701).
( ) 1
1
415
nn
n
¥+
=
-å
lim nnb
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{ }nb
Section 11.5 Suggested reading HW (not collected) MATH 1152Q Alternating Series Notes
Page 6 of 6
TASK 8. Example:
Determine whether converges.
Step (ii): Calculate
Step (i): Estimate (in your head or scratch paper) whether is decreasing. If you believe it is decreasing, verify by:
• computing a derivative (follow Sec 11.5 Example 3 page 734), • or directly (follow Sec 11.1 Example 13 page 701).
( ) 12
1
21n
n
n n
¥+
=
-å
lim nnb
®¥=
{ }nb
