Nth Term Test: The series diverge because the limit as goes to infinity is zero. P-Series Test: The summation converges since. Root Test: Since the limit as approaches to infinity is zero, the series is convergent. Divergence Test: Since limit of the series approaches zero, the series must converge. The series is a harmonic series. The Nth term test and the Divergent test may not be used to determine whether this series converges, since this is a special case.
The root test also does not apply in this scenario. According the the P-series Test, must converge only if. Therefore this could be a valid test, but a wrong definition as the answer choice since the series diverge for.
Given just the harmonic series, we would state that the series diverges. However, we are given the alternating harmonic series. To determine whether this series will converge or diverge, we must use the Alternating Series test. The test states that for a given series where or where for all n, if and is a decreasing sequence, then is convergent. 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.
Next, we must determine if is a decreasing sequence. No the series does not converge. The given problem is the harmonic series, which diverges to infinity. The series converges. The given problem is the alternating harmonic series, which converges by the alternating series test.
Only the Integral Test will work on the Harmonic Series,. Since , the Limit Test for Divergence fails. The Ratio Test and the Root Test will always yield the same conclusion, so if one test fails, the both fail and vise versa. Since the result of the limit is , both tests fail. You are then asked to determine if the series converges, or diverges. For what values of p would this series be convergent?
The series converges, if , and diverges, if. If you've found an issue with this question, please let us know. With the help of the community we can continue to improve our educational resources. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors. Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.
Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney. 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.
The number in the denominator would be large, but the numerator would be larger. You can write the digits down sort of , but they don't have a number associated with them this way. 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.
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