Use the geometric series test to find whether the given series converges or diverges. If it converges, find its sum. 3)

Question

_
Q
k = 1
7
k + 1
9
k - 1

Answer 1

To determine if the given series converges or diverges, we need to check if the common ratio, r, falls within the range of -1 < r < 1.

The given series is:
∑ (7k+1 / 9k-1)

Let's find the value of r:
r = (7(k+1) / 9(k-1)) / (7k / 9k) = (7k + 7) / (9k - 9) * (9k) / (7k)
r = (7k + 7) / (7k - 9)

Now let's check if -1 < r < 1:
-1 < (7k + 7) / (7k - 9) < 1

Next, we can solve for the lower limit:
-1 < (7k + 7) / (7k - 9)
-7k + 9 < 7k + 7
-14 < 14k
-1 < k

So, the lower limit for the common ratio k is k > -1.

Now let's solve for the upper limit:
(7k + 7) / (7k - 9) < 1
7k + 7 < 7k - 9
16 < -9

This inequality is not possible, so there is no upper limit for the common ratio k.

Therefore, the given series diverges.