If the limit as n goes to infinity of the summation from n equals 1 to n of a sub k exists and has a finite value, the infinite series the summation from n equals 1 to n of a sub k is said to be (4 points)
A) unbounded
B) convergent
C) increasing
D) divergent
convergent
If the sum has a finite limit, it converges (to that limit!)
better review these terms some more
To determine the answer to this question, we need to understand the concepts of convergence and divergence in infinite series.
An infinite series is said to be convergent if the limit of the partial sums exists and has a finite value. In other words, if the sum of all the terms in the series approaches a specific value as we add more and more terms, then the series converges. On the other hand, if the partial sums do not approach a specific value and keep growing without bound, the series is said to be divergent.
In this case, since we are given that the limit as n goes to infinity of the summation from n equals 1 to n of a sub k exists and has a finite value, it implies that the partial sums of the series have a well-defined limit. Therefore, the series is convergent. Hence, the correct answer is:
B) convergent