Integral test

1. Integral test • Let {an} be a sequence of positive terms and f(x) be a continuous function such that • f(n) = an for all n > 0, and • f(x) is decreasing on [0, ) • f(x)  0 as n , then

2. 1 2 3 4 5 6 7 8 9

3. Height of rectangle = f(1) = a1 Width of rectangle = 1  Area of rectangle = a1 area = a1 1 2 3 4 5 6 7 8 9

4. Height of rectangle = f(2) = a2 Width of rectangle = 1  Area of rectangle = a2 area = a1 area = a2 1 2 3 4 5 6 7 8 9

5. area = a1 area = a2 area = a3 1 2 3 4 5 6 7 8 9

6. area = a1 area = a2 area = a3 1 2 3 4 5 6 7 8 9

7. Integral test Warning: Even if both the series and the improper integral are convergent, they will not converge to the same value!

8. Integral test Estimation by finite partial sums. In general, even if the series is convergent, its sum cannot be computed by a formula. In that case we can only get an approximation by adding the first “few” terms. How good is our approximation?

9. Integral test Theorem Suppose that is the exact value of the convergent series, and is the partial sum of the first k terms, then

10. Comparison test • Let {an} and {bn} be sequences of positive terms such that 0  an  bn , • If bn isconvergent, then so is an • If an is divergent, then so is bn

11. Limit Comparison test Let {an} and {bn} be sequences of positive numbers, then anandbnbehave the same way.

12. Alternating Series A series en is said to be alternating if each pair of adjacent terms have opposite signs, e.g. is an alternating series. We normally write an alternating series in the form (-1)nbn or (-1)n+1bnwhere bnis assumed to be positive for all n > 0.

13. Alternate Series Test • An alternating series (-1)nbn is convergent if • {bn} is eventually decreasing and • limnbn = 0. Example: The alternating harmonic series is convergent.

14. Absolute and Conditional Convergence A series an is said to be absolutely convergent if  |an| is convergent. (remark: an absolutely convergent series is of course, convergent.) A series is said to be conditionally convergent if an is convergent but  |an| is divergent (i.e. the sum is ) Example: the alternating harmonic series is conditionally convergent.

15. Special properties: • If an is an absolutely convergent series, then any rearrangement of it will also converge to the same sum. • If an is conditionally convergent, then for any real number M, we can find an rearrangement bn of an such that bn converges to M.

16. Ratio Test Let an be a series of real numbers, and suppose that

17. Root Test Let an be a series of real numbers, and suppose that

18. Remark If either one of the ratio test or the root test fails for an , then so will the other. The ratio test is however, more convenient for series whose n-th term involves n!.

19. Delicate ratio test Let an be a series of real numbers.

20. Delicate root test Let an be a series of real numbers.