1. [-/2 Points] DETAILS MY NOTES SCALCET9 11.4.001. Suppose \\ a and > b are series with positive terms and b is known to be convergent. (a) If a > b, for all n, what can you say about a,? Why? O We cannot say anything about Can a converges if and only if an $ 2b. a converges if and only if a s 4b. a converges by the Comparison Test. a diverges by the Comparison Test. (b) If a b for all n, what can you say about a,? Why? a converges by the Comparison Test. a converges if and only if 2a 2 br. a diverges by the Comparison Test. O We cannot say anything about an a converges if and only if na z br. (b) If an 5n 1, Therefore Z is by the Comparison Test. 5n2 +2n + 8 15. [-/3 Points] SCALCET9 11.4.AE.002. PRACTICE Example Video Example o Test the series Z In(5K) for convergence or divergence. 2 k=1 Solution We used the Integral Test to test this series in this example, but we can also test it by comparing it with the harmonic series. Observe that In(5k) > 1 for k > 3/5 and so we have the following. 1 k= 3/5 2k / In(5k) 2k @ We know Z 1 s divergent (it's a pseries with p = I: ) Thus the given series is by the Comparison Test. (2k) k=1 Need Help? 1. [2/3 Points] DETAILS MY NOTES SCALCET9 11.3.003. Use the Integral Test to determine whether the series is convergent or divergent. o0 > n=1 Evaluate the following integral. oo / x " 8dx 1 Since the integral " finite, the series is v Need Help? 2. [2/3 Points] DETAILS MY NOTES SCALCET9 11.3.005. Use the Integral Test to determine whether the series is convergent or divergent. 00 E 4 7n - 1 n = 1 Evaluate the following integral. 4 dx 1 7x - 1 Since the integral is not V finite, the series is | divergent v Need Help? Read It3. [2/3 Points] DETAILS MY NOTES SCALCET9 11.3.007. Use the integral test to determine whether the series is convergent or divergent. o 2 n Z nd+1 n=2 Evaluate the following integral. w0 2 / X dx 2 x2+1 Since the integral finite, the series is & Need Help? 4. [2/3 Points] DETAILS MY NOTES SCALCET9 11.3.008. Use the Integral Test to determine whether the series is convergent or divergent. n = 1 Evaluate the following integral. x8e-x dx Since the integral | is V finite, the series is | convergent v Need Help? Read It9. [3/4 Points] DETAILS MY NOTES SCALCET9 11.3.019.EP. Consider the following series. o0 Z 1 2 no1 Tt 25 1 Does the function f(x) = - satisfy the conditions of the Integral Test? X4 Evaluate the following integral. If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.) o 1 Zidx i X 0, then the function f(x) = - is clearly continuous, positive, and decreasing on [1, ). Previously, we found that X! X o 1 . ; . dx converges if p > and diverges if p \\:I and diverges if 0