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SUMMARY OF TESTS
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SUMMARY OF TESTS Special Series: Series Tests Geometric Series
Harmonic Series Telescoping Series Alter Harmonic p-series Alternating p-series Divergence Test Integral Test Comparison Test Limit Compar Test Ratio Test Root Test Alter Series Test
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STRATEGY FOR TESTING SERIES
PART-1: Series with positive terms STRATEGY FOR TESTING SERIES Divg Test factorial: ratio test comp+lim comp power of n: root test easy to integrate: integral test similar to geometric: try comp+lim comp Similar to p-series: try comp+lim comp PART-2: Alternating Series Study (use PART-1) convg divg use alt.-Test AC convg divg CC PART-3: Series with some negative terms REMARK: For multiple-choice-question: before you start read the alternatives first. It guides you to which tests you need to use. Study (use PART-1) convg divg AC
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PART-1: Series with positive terms 151 121
Divg Test factorial: ratio test comp+lim comp power of n: root test easy to integrate: integral test similar to geometric: try comp+lim comp Similar to p-series: try comp+lim comp 141 122 Divg Test Integral Tst Compar Tst LimComp Ratio Tst Root Tst AltSer Tst
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PART-1: Series with positive terms 151 121
Divg Test factorial: ratio test comp+lim comp power of n: root test easy to integrate: integral test similar to geometric: try comp+lim comp Similar to p-series: try comp+lim comp 141 122 Divg Test Integral Tst Compar Tst LimComp Ratio Tst Root Tst AltSer Tst
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Final-121
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Final-121
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Final-122
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Final-121
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Final-131
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Final-131
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Final-131
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Final-131
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S = Sn + Rn S ~ Sn ~ ESTIMATING THE SUM OF A SERIES
Remark: We know how to find the sum of geometric series and telescoping series Example: ESTIMATING THE SUM OF A SERIES Estimate the sum Euler found the sum of the p-series with p=4 S = Sn Rn S ~ Sn ~ for sufficient large n Sn Rn 8.2323e-02 1.9823e-02 7.4776e-03 3.5713e-03 1.9713e-03 1.1997e-03 7.8321e-04 5.3907e-04 3.8665e-04 2.8665e-04 2.1835e-04 n 1 2 3 4 5 6 7 8 9 10 11 S = S = How many terms are required to ensure that the sum is accurate to within 0.001? Example:
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4) How many terms are needed within error
PART-1: Series with positive terms PART-2: Alternating Series Bounds for the Remainder in the Integral Test Alternating Series Estimation Theorem Convergent by integral test approximates the sum S of the series with an error whose absolute value is less than the absolute value of the first unused term good approx Error (how good) The theorem says that for series that satisfy the conditions of the Alternating Series Test, the size of the error is smaller than
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4) How many terms are needed within error
PART-1: Series with positive terms PART-2: Alternating Series
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ALTERNATING SERIES
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ALTERNATING SERIES TERM-102
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4-types Types of questions in the exam
1) Determine whether convg or divg AC,CC 2) Find the sum s 4-types 5) Partial sums and their properties 4) How many terms are needed within error
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Extra Problems
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Final-122
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Final-122
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Final-122
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