Please state true or false with explanation if true and counterexample if false.

Question

Please state true or false with explanation if true and counterexample if false.

No points given if no explanation or counterexample given.

Thank you in advance.

Explain why or why not: Determine whether the following statements are true or false and give an explanation if true or counterexample if false. A series that converges must converge absolutely. A series that converges absolutely must converge. A series that converges conditionally must converge. If ak diverges, then |ak| diverge If ak converges conditionally, then |ak|diverges. If ak > 0 and ak converges, then a2k converges. In general, the second and third degree Taylor polynomials, p2 and p3 respectively, have 2 terms in common.

Explanation / Answer

a. false. (-1)^n*1/n converges, but not absolutely

b. true. absolute convergence guarantees the series goes to 0, which is all that is required to show convergence of an alternating series.

c. true. this is the definition of conditional convergence.

d. true. the absolute value will be at least as big so we can use the direct comparison test.

e. true. this is the definition of conditional convergence.

f. true. for a series to converge, the nth term must go to 0. thus at some point the terms are all less than 1. at this point the square of a number is less than the number itself so we can apply the direct ocmparison test.

g. false. 1/n^2 converges, but 1/n diverges

h. true. the error bounds get smaller as n goes to infinity.

i. they should have three terms in common, so if this means at least two terms then this is true.

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