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

2. 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:

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

5. ALTERNATING SERIES

6. ALTERNATING SERIES
TERM-102

7. 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