Question

Find the sum of the series and identify the type of sequence (convergent or divergent):

∑ from n = 1 to [infinity] of ((-1)^(n-1) * (7 / 6^(n-1)))

Find the sum of the series and identify the type of sequence (convergent or divergent):
∑ from n = 1 to [infinity] of (2 / ((4n-3)(4n+1)))

Find the sum of the series and identify the type of sequence (convergent or divergent):
∑ from n = 1 to [infinity] of (1 / (9n² + 3n - 2))

Answer 1

To determine the convergence and find the sum of a series, one must identify the series type and apply appropriate formulas or tests. For instance, a geometric series with a common ratio less than one is convergent, and its sum can be calculated with the geometric series sum formula. Other series may require different methods like telescoping or convergence tests.

Step-by-step explanation:

To find the sum of a series and determine whether it is convergent or divergent, there are a few steps to follow:

1. Identify the type of series (geometric, telescoping, etc.).
2. Apply appropriate formulas or tests to determine if the series converges.
3. If convergent, use the sum formula to find the total sum. If divergent, state that the series does not have a sum.

For example, the first series given is a geometric series with a common ratio of -7/6. Since the absolute value of the common ratio is less than 1, the series is convergent, and we can use the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio, to find its sum.

The second and third series require specific techniques to identify convergence and calculate the sum, such as telescoping series methods or convergence tests like the p-test or comparison test.