ESTIMATING THE SUM OF A SERIES 11.3 Integral Test 11.5 Alternating Test

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 1.082323233711138 Sn 1.000000000000000 1.062500000000000 1.074845679012346 1.078751929012346 1.080351929012346 1.081123533950617 1.081540027078481 1.081784167703481 1.081936583493757 1.082036583493757 1.082104884839293 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 = 1.082323233711138 1.082322905344473 1.082323192355929 1.082323233378306 9000 1.082323233710697 S = 1.082323233711138 How many terms are required to ensure that the sum is accurate to within 0.001? Example:

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

4) How many terms are needed within error PART-1: Series with positive terms PART-2: Alternating Series

ALTERNATING SERIES

ALTERNATING SERIES TERM-102

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