Chapter 7: Q. 8 (page 655)
Some Convergent Sequences Involving Exponents: For any real number p > 0, the following sequences converge. Fill in each blank with the appropriate value.
Short Answer
The required answer is
Chapter 7: Q. 8 (page 655)
Some Convergent Sequences Involving Exponents: For any real number p > 0, the following sequences converge. Fill in each blank with the appropriate value.
The required answer is
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Improper Integrals: Determine whether the following improper integrals converge or diverge.
Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.
Let be any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to . (Hint: Argue that if you add up some finite number of the terms of , the sum will be greater than . Then argue that, by adding in some other finite number of the terms of
, you can get the sum to be less than . By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to .)
For each series in Exercises 44–47, do each of the following:
(a) Use the integral test to show that the series converges.
(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.
(c) Use Theorem 7.31 to find a bound on the tenth remainder,.
(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.
(e) Find the smallest value of n so that
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