Final answer:
The values of b for which the series converges are b > 1.
Step-by-step explanation:
To find the values of b for which the series converges, we need to use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges. Applying the ratio test to the series bln(n), we have:
lim(n->∞) |(bn+1ln(n+1)) / (bln(n))| < 1
Simplifying the limit and incorporating the properties of logarithms, we get:
lim(n->∞) ln((n+1)/n) < 1/b
Taking the natural logarithm of both sides and using the limit definition of the natural logarithm, we have:
ln(lim(n->∞) (n+1)/n) < ln(1/b)
Simplifying further, we get:
ln(1) < ln(1/b)
Since ln(1) = 0, the inequality becomes: 0 < ln(1/b)
Therefore, the values of b for which the series converges are b > 1.