how do I know if the series ln(n)/(2n) is DV or CV?
Do you mean divergent or convergent?
Yes. I found that the limit of the sequence goes to 0 and that it is a positive term series. Now what do I do?
ln(n)/n > 1/n and the series 1/n diverges.
To determine whether the series ln(n)/(2n) is divergent (DV) or convergent (CV), we can apply the limit comparison test.
Step 1: Choose a known convergent series with positive terms that has a similar behavior to the given series. A suitable candidate for comparison is the harmonic series: 1/n.
Step 2: Take the limit of the ratio of the terms of the two series as n approaches infinity:
lim(n→∞) ln(n)/(2n) / (1/n)
Step 3: Simplify the expression by multiplying the numerator and the denominator by n:
lim(n→∞) ln(n)/(2n) * (n/n) / (1/n)
This yields:
lim(n→∞) ln(n)/2 / 1
Step 4: Simplify further:
lim(n→∞) ln(n)/2
Step 5: Apply the l'Hôpital's rule to evaluate the limit:
lim(n→∞) 1/n / 2 = 0
Step 6: Compare the result of the limit to determine the convergence of the given series. Since the limit is not equal to zero, the series ln(n)/(2n) is divergent.
Therefore, the series ln(n)/(2n) is determined to be divergent (DV).