Chapter 7: Q. 7 (page 624)
Let f(x) be a function that is continuous, positive, and decreasing on the interval such that , What can the integral tells us about the series ?
Short Answer
The series is divergent.
Chapter 7: Q. 7 (page 624)
Let f(x) be a function that is continuous, positive, and decreasing on the interval such that , What can the integral tells us about the series ?
The series is divergent.
All the tools & learning materials you need for study success - in one app.
Get started for freeLet 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 a convergent series satisfying the conditions of the integral test, why is every remainder positive? How can be used along with the term from the sequence of partial sums to understand the quality of the approximation ?
Determine whether the series converges or diverges. Give the sum of the convergent series.
Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.
Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.
What do you think about this solution?
We value your feedback to improve our textbook solutions.