MATH 126 Calculus II Learning Goal Activity 10 2 10 3 Name Due Date October 13 2023 11 59pm On Canvas LEARNING GOALS to determine whether a given series is a geometric series by nding a common ratio to determine whether a geometric series converges or diverges and to nd the sum of a convergent geometric series using its common ratio 1 Determine whether the following series converge or diverge If a series converges then nd its sum a 4 3 16 9 64 27 256 81 cid 18 4 cid 19 n 5 cid 88 n 1 b 3 cid 88 n 1 c 22n 3n 5n 1 LEARNING GOAL in nite series converges or diverges in an abstract case and telescoping series situation to nd and use an in nite series s sequence of partial sums to determine whether an 2 Given the following series converges nd the sum of the series cid 88 n 1 3 n n 3 3 Suppose the following are true S1 a1 S2 a1 a2 S3 a1 a2 a3 S4 a1 a2 a3 a4 4 25 16 125 64 625 Determine whether an converges or diverges by rst nding the general term of its sequence SN of partial sums If the series converges then nd its sum 1 5 cid 88 n 1 2 LEARNING GOAL whether the limit at in nity of the general term of the series is 0 to use the nth Term Divergence Test to show a given series diverges by checking 4 Suppose we need to determine whether each of the following series converge or diverge Justify whether the nth Term Divergence Test can be used to make a conclusion in the space below each series If you can make a conclusion then write that conclusion cid 88 n 1 n 5n a cid 88 k 1 1 k6 b c sin n n cid 88 n 1 cid 88 k 2 cid 88 n 2 d arctan k e n4 2n4 1 3 LEARNING GOAL to use the Integral Test to show a given series converges or diverges by verify ing the three pre requisite conditions of the Integral Test are met and then evaluating the improper integral corresponding to the given in nite series 5 Use the Integral Test to determine whether the following series converge or diverge check criteria needed cid 88 n 2 a ne n2 cid 88 k 1 b 1 k2 1 cid 88 n 2 c 1 n ln n 3 4 cid 88 k 3 d ln k k cos cid 0 1 n cid 1 n2 cid 88 n 3 e LEARNING GOAL p value is 1 or 1 to determine whether a p series converges or diverges by determining whether its 6 Determine whether each of the following p series converges or diverges using Theorem 3 Convergence of p series In each case explain your answer using the value of p cid 88 n 1 1 n 4 cid 88 n 1 1 n8 cid 88 n 1 1 n1 e cid 88 n 1 1 n6 5 5 LEARNING GOAL to use the Direct Comparison Test or the Limit Comparison Test to show a given series converges or diverges by selecting an appropriate comparison series verifying that both the comparison series and given series have positive terms determining whether the comparison series converges or diverges and comparing the two series general terms either directly or via a limit 7 Use the Direct Comparison Test or the Limit Comparison Test to determine whether the following series converge or diverge cid 88 n 2 a n 1 n5 n2 cid 88 n 2 b n 3n 7n n cid 88 n 1 c n 1 n6 6 cid 88 k 2 d k2 k k8 2k2 1 cid 88 k 1 e 4 3k 1 cid 88 n 5 f n2 1 8n ln n 5n cos n n2 7