The comparison tests

1. The comparison tests • Theorem Suppose that and are series with positive terms, then (i) If is convergent and for all n, then is also convergent. (ii) If is divergent and for all n, then is also divergent. • Ex. Determine whether converges. • Sol. So the series converges.

2. The limit comparison test • Theorem Suppose that and are series with positive terms. Suppose Then (i) when c is a finite number and c>0, then either both series converge or both diverge. (ii) when c=0, then the convergence of implies the convergence of (iii) when then the divergence of implies the divergence of

3. Example • Ex. Determine whether the following series converges. • Sol. (1) diverge. choose then (2) diverge. take then (3) converge for p>1 and diverge for take then

4. Question • Ex. Determine whether the series converges or diverges. • Sol.

5. Alternating series • An alternating series is a series whose terms are alternatively positive and negative. For example, • The n-th term of an alternating series is of the form where is a positive number.

6. The alternating series test • Theorem If the alternating series satisfies (i) for all n (ii) Then the alternating series is convergent. • Ex. The alternating harmonic series is convergent.

7. Example • Ex. Determine whether the following series converges. • Sol. (1) converge (2) converge • Question.

8. Absolute convergence • Aseries is called absolutely convergent if the series of absolute values is convergent. • For example, the series is absolutely convergent while the alternating harmonic series is not. • Aseries is called conditionally convergent if it is convergent but not absolutely convergent. • Theorem. If a series is absolutely convergent, then it is convergent.

9. Example • Ex. Determine whether the following series is convergent. • Sol. (1) absolutely convergent (2) conditionally convergent

10. The ratio test • The ratio test (1) If then is absolutely convergent. (2) If or then diverges. (3) If the ratio test is inconclusive: that is, no conclusion can be drawn about the convergence of

11. Example • Ex. Test the convergence of the series • Sol. (1) convergent (2) convergent for divergent for

12. The root test • The root test (1) If then is absolutely convergent. (2) If or then diverges. (3) If the root test is inconclusive.

13. Example • Ex. Test the convergence of the series • Sol. convergent for divergent for

14. Rearrangements • If we rearrange the order of the term in a finite sum, then of course the value of the sum remains unchanged. But this is not the case for an infinite series. • By a rearrangement of an infinite series we mean a series obtained by simply changing the order of the terms. • It turns out that: if is an absolutely convergent series with sum , then any rearrangement of has the same sum . • However, any conditionally convergent series can be rearranged to give a different sum.

15. Example • Ex. Consider the alternating harmonic series Multiplying this series by we get or Adding these two series, we obtain

16. Strategy for testing series • If we can see at a glance that then divergence • If a series is similar to a p-series, such as an algebraic form, or a form containing factorial, then use comparison test. • For an alternating series, use alternating series test.

17. Strategy for testing series • If n-th powers appear in the series, use root test. • If f decreasing and positive, use integral test. • Sol. (1) diverge (2) converge (3) diverge (4) converge

18. Power series • A power series is a series of the form where x is a variable and are constants called coefficients of series. • For each fixed x, the power series is a usual series. We can test for convergence or divergence. • A power series may converge for some values of x and diverge for other values of x. So the sum of the series is a function

19. Power series • For example, the power series converges to when • More generally, A series of the form is called a power series in (x-a) or a power series centered at a or a power series about a.

20. Example • Ex. For what values of x is the power series convergent? • Sol. By ratio test, the power series diverges for all and only converges when x=0.

21. Homework 24 • Section 11.4: 24, 31, 32, 42, 46 • Section 11.5: 14, 34 • Section 11.6: 5, 13, 23 • Section 11.7: 7, 8, 10, 15, 36