Why does harmonic series diverge

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Hanley Rd, Suite St. Louis, MO Subject optional. Email address: Your name:. Example Question 1 : Harmonic Series. Possible Answers:. Correct answer:. Explanation : A -series is a series of the form , and the Harmonic Series is. Report an Error. Example Question : Calculus Ii. Integral Test: The improper integral determines that the harmonic series diverge.

Correct answer: Integral Test: The improper integral determines that the harmonic series diverge. Explanation : The series is a harmonic series. This leaves us with the Integral Test.

Since the improper integral diverges, so does the series. Determine whether the following series converges or diverges:. Possible Answers: The series absolutely converges. The series may absolutely converge, diverge, or conditionally converge. Correct answer: The series absolutely converges. Explanation : Given just the harmonic series, we would state that the series diverges. First, we must evaluate the limit of as n approaches infinity: The limit equals zero because the numerator of the fraction equals zero as n approaches infinity.

Because both parts of the test passed, the series is absolutely convergent. Consider the alternating series. Which of the following tests for convergence is NOT conclusive? Possible Answers: The alternating series test. Correct answer: The limit test for divergence. Explanation : Let be the nth summand in the series. The limit test for divergence states that implies that the series diverges. However, , so the test is inconclusive. Does the following series converge?

Possible Answers: Yes. Correct answer: No. Explanation : No the series does not converge. Example Question 2 : Harmonic Series.

Possible Answers: Cannot be determined. Correct answer: Yes. Explanation : The series converges. Example Question 3 : Harmonic Series. Possible Answers: The Root Test. Correct answer: The Integral Test. To use the Integral Test, we evaluate , which shows that the series diverges. For the Ratio Test,. Example Question 4 : Harmonic Series. Let's say you are given harmonic series in the following form: ; You are then asked to determine if the series converges, or diverges.

Explanation : The given series is called generalized harmonic series. Copyright Notice. View Calculus 2 Tutors. Joseph Certified Tutor. Samuel Certified Tutor. Chase Certified Tutor.

A real number is in fact often defined as the limit of a converging sequence, so a sequence that gets larger and larger can't be a real number by definition. Whatever answer you give, I can go higher, so the answer doesn't exist.

The so called Divergence test sais that if the first happens then the second must hold, but the converse is not true. And here is the reason why:. And anything can happen in this case. For this series, what happens is that more and more terms will agree on the first "few" digits, but then KABOOM you get a term with new digits.. Sign up to join this community. The best answers are voted up and rise to the top.

Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why do we say the harmonic series is divergent? Asked 9 years, 3 months ago. Active 9 years, 2 months ago. Viewed 27k times. Nick Anderegg Nick Anderegg 2 2 gold badges 5 5 silver badges 10 10 bronze badges.

This is a serious question. But in a sum, the value may be getting closer to zero, but the running total still continues to get bigger. Show 9 more comments. Active Oldest Votes. And so on. That end part is a perfect explanation. That sums up what I was thinking, the harmonic series really, really, really wants to converge if we're anthropomorphizing numbers now , but it can't quite get there. I get it now. Add a comment. Unfortunately, that is not true.

It would simplify a lot of math if it was true. Have you seen the proof that it diverges? Old John