Determine whether the series sum a_n an converges or diverges: Use the Alternating Series Test. sum_{n=2}^infty(-1)^nfrac{n}{ln(n)}

Cheyanne Leigh

Cheyanne Leigh

Answered question


Determine whether the series an an converges or diverges: Use the Alternating Series Test.

Answer & Explanation



Skilled2021-03-13Added 98 answers

Consider the series n=2(1)nnln(n)
Take an=(1)nnln(n) and bn=nln(n)
The Alternating series test is stated below:
Suppose the series an and either an=(1)nbn or an=(1)n+1bn where bn0 for all n. Then if,
2. {bn} is a decreasing sequence
The series is convergent.
Check the first condition for series convergent.
The value of the limit is in the indeterminate form.
Apply LHopitals rule to find the limit as follows.
Observe that, the limit of the sequence goes to infinity as x goes to infinity. Thus, the series does not converges.
The divergence test states that, "If limnan0 then the series an will diverge".
Since limnbn0, the limit of the function an=(1)nnln(n) also not equal to zero as x goes to zero. That is, limnan0.
By divergence test, it is concluded that the alternating series n=2(1)nnln(n) diverges.
Jeffrey Jordon

Jeffrey Jordon

Expert2021-12-27Added 2607 answers

Answer is given below (on video)

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?