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Magazine Chapter 7: Sequences and Series
Q 70.
Find the values of x for which the series converges.
Q. 70
Determine whether the sequences in Exercises 63鈥�74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
Q. 70
Let and be convergent sequences with and as and let be a constant. Prove the indicated basic limit rules from Theorem 7.11. You may wish to model your proofs on the proofs of the analogous statements from Section 1.5.
Prove that if , then .
Q. 70
Prove the ratio test for absolute convergence. That is, let be a series with nonzero terms, and let localid="1650855913677"
(a) Show that if 蚁 < 1, the series converges absolutely.
(b) Show that if 蚁 > 1, the series diverges.
(c) Show that the test fails when 蚁 = 1, by finding a convergent series such as
Changing the order of the summands in a conditionally convergent series can change the value of the sum. We showed this earlier in the section for the alternating harmonic series
Q. 71
Let be a sequence. Prove the indicated limit rules from Theorem 7.12. You may wish to model your proofs on the proofs of the analogous statements from Section 1.5.
Prove that if and , then .
Q. 71
Changing the order of the summands in a conditionally convergent series can change the value of the sum. We showed this earlier in the section for the alternating harmonic series
Q. 71
Determine whether the sequences in Exercises 63鈥�74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
Q. 71
Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.
Q. 72
Prove that if and is a function that is continuous at , then .
Q. 72
Determine whether the sequences in Exercises 63鈥�74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
