Question

Exercise. Select all of the following series that converge: the series represented by f(−2), the series represented by f(0), the series represented byf(2), k 5 . the series represented by f(4), the series represented by f(5), the series represented by f(7).

a) f(−2) and f(0)
b) f(2) and f(4)
c) k^5 and f(7)
d) f(2) and f(5)

Answer 1

The convergence of a series is determined by the behavior of its terms as the index approaches infinity.The series that converge are f(−2) and f(0). Thus, the correct option is a) f(−2) and f(0).

Step-by-step explanation:

In this case, without specific information about the function f, we can analyze the given options.

For f(−2) and f(0), the series converges since the indices are fixed at −2 and 0. These are individual terms and can be considered as constant values, resulting in convergent series.

On the other hand, for f(2), f(4), f(5), and f(7), the behavior of the series is not explicitly mentioned. Without additional information, we cannot determine whether these series converge or diverge.

It's important to note that the series represented by k^5 is not related to f(x) and is, therefore, unrelated to the convergence of the given series.

In conclusion, the series that converge are f(−2) and f(0), as they represent fixed values and do not involve the sum of an infinite number of terms. The convergence of the other series depends on the specific behavior of the function f, which is not provided in the given options.

Therefore, the correct answer is a) f(−2) and f(0).