Question

−8/11+64/121−512/1331+4096/14641+⋯ Determine whether the series converges, and If it converges, determine its value. Enter the value if the series converges, or enter DNE a it diverges: Problem 15. (1 point) Consider the series −8/11+64/121−512/1331+4096/11611+… Determine whether the series converges, and if it converges, determine its value. Enter the value if the serios converges, or enter DNE if it diverges

Answer 1

The series converges, and its sum is -8/13. It is a geometric series with each term of the form ((-2)^n)/(11^n) and a common ratio of -2/11.

Step-by-step explanation:

The series presented is -8/11 + 64/121 - 512/1331 + 4096/14641 + … where the pattern appears to be of each term being of the form ((-2)^n)/(11^n). To evaluate whether this series converges, we can observe that it resembles a geometric series with the common ratio r = (-2/11). Since the absolute value of the common ratio is less than 1 (|r| < 1), the series converges.

A convergent geometric series has the sum S given by S = a/(1 - r), where 'a' is the first term. Here, 'a' is -8/11 and 'r' is -2/11. Substituting these values into the formula, we get S = (-8/11) / (1 - (-2/11)) = (-8/11) / (13/11), which simplifies to S = -8/13.