微积分教学课件chapter8.4.ppt

Proof of The Integral Test: Example: The Comparison Test Example: Example: Example: Example: Example: Example: * 8.4 Series of Nonnegative Terms A series with nonnegative terms is convergent iff the sequences of its partial sums is bounded. Series of Nonnegative Terms Corollary of Theorem 5 For example Converges Diverges Theorem: (4) (5) Example Test the series for convergence or divergence. solution Because and when so the function is continuous, positive,and decreating on . We use the integral test: . Thus, is convergent integral and so,by the integral test,the series is convergent. Converges Diverges Note: Note: proof (i) let Since both series have positive terms, the sequences and are increasing. If is convergent then for all n, Since we have for all n. This means that is incrasing and bounded and therefore converges by the Monotonic Sequence Theorem.thus, converges. (ii) If is convergent then, Since we have for all n. Thus, Therefore diverges. Most of the time we use a p-series or a geometric series for the purpose of comparison Example Determine whether the series converges Or diverges: Solution diverges diverges converges converges Example Determine whether the series converges Or diverges: Solution: converges converges Solution: and converges Solution: and Theorem: then: and So the given series converges by the limit comparison test. So the given series converges by the limit comparison test. The Ratio Test Let be a series with positiv terms and suppose that Then the series converges if the series deverges if or is infinite the test is inconclusive if Proof (a) Let when Since Thus That is *