Question

determine whether the series is convergent or divergent. Σn=1 [infinity] (-6)^n-1/7^n. O convergent O divergent. if it is convergent, find its sum. (if the quantity diverges, enter diverges.)

Answer 1

divergent

Explanation:

The given series is:

Σn=1 to infinity (-6)^(n-1) / 7^n

To determine if the series is convergent or divergent, we can use the ratio test, which states that if the absolute value of the ratio of consecutive terms in a series converges to a value less than 1, then the series converges; if the ratio converges to a value greater than 1 or does not converge, then the series diverges.

Let's apply the ratio test to the given series:

|(-6)^(n-1) / 7^n| / |(-6)^n / 7^(n+1)|

= |(-6)^(n-1)| / 7^n * |7^(n+1)| / |(-6)^n|

= |-6|^(n-1) / 7^n * |7|^(n+1) / |-6|^n (taking absolute values and rearranging)

= 6^(n-1) / 7^n * 7^(n+1) / 6^n (simplifying absolute values)

= (6/7) * (7/6)^n

As n approaches infinity, (7/6)^n approaches infinity since 7/6 is greater than 1. Therefore, the ratio of consecutive terms does not converge to a value less than 1, which means the series diverges.

So, the given series is divergent.