Question

Which of the following series can be used to determine the convergence of the series summation from k equals 0 to infinity of a fraction with the square root of quantity k to the eighth power minus k cubed plus 4 times k minus 7 end quantity as the numerator and 5 times the quantity 3 minus 6 times k plus 3 times k to the sixth power end quantity squared as the denominator question mark

Answer 1

To determine its convergence, we can use the comparison test. We consider two series for comparison:

Series 1: $\sum_{k=0}^\infty \frac{k^8}{5(3-6k+3k^6)^2}$

Series 2: $\sum_{k=0}^\infty \frac{k^8 + k^3 + 4k}{5(3-6k+3k^6)^2}$

We notice that Series 2 is always greater than or equal to Series 1.

Next, we use the p-test, which states that if the ratio of consecutive terms in a series approaches a value less than 1, then the series converges. For Series 1, the ratio of consecutive terms approaches 1, which means Series 1 diverges.

Since Series 1, which is smaller than Series 2, diverges, we can conclude that Series 2 also diverges.

Therefore, based on the comparison test, the given series also diverges.

Explanation: