Question

determine all values of p for which the series x[infinity] n = 2 (−1)n−1 (ln(n))p 2n is convergent, expressing your answer in interval notation. 1. (−[infinity], 0) 2. (−[infinity], [infinity]) 3. [0, [infinity]) 4. (0, [infinity]) 5. p = {0}

Answer 1

The values of p that make the series convergent are in the interval (-∞, 0).

Step-by-step explanation:

To determine the values of p for which the series xn = 2(−1)n−1(ln(n))p2n is convergent, we need to find the values of p that will ensure the series converges. For a series to converge, the general term of the series must approach zero as n approaches infinity. Therefore, the exponent of ln(n) must be less than zero, i.e., p < 0. Hence, the values of p that make the series convergent are in the interval (-∞, 0) which is option 1.

Answer 2

The series converges for all values of p, which can be shown using the Ratio Test, resulting in a conclusion that every value of p satisfies the ratio being less than 1. The interval of convergence in notation is (-∞, ∞).

Step-by-step explanation:

To determine all values of p for which the series ∑ from n = 2 to infinity of (−1)n−1 (ln(n))p / 2n is convergent, we can apply the Ratio Test, because the series involves both alternation and the natural logarithm, which can complicate direct comparison or integral tests. The Ratio Test states that a series ∑ an converges absolutely if the limit as n approaches infinity of |an+1 / an| is less than 1.

Let an = (−1)n−1 (ln(n))p / 2n. We find the ratio |an+1 / an| and take the limit as n approaches infinity. If this limit is less than 1 for certain values of p, those are the values for which the original series converges.

In order to do this, we would calculate the following limit:

limn→∞ |an+1/an|

= limn→∞ |(−1)n (ln(n+1))p / 2n+1 / ((−1)n−1 (ln(n))p / 2n)|

The absolute value signs negate the alternating factor, and the ratio of the powers of 2 simplifies to 1/2. Thus, we're left with analyzing the limit involving natural logarithms. As n approaches infinity, ln(n+1)/ln(n) approaches 1, so the entire limit approaches (1/2) for any value of p. This is always less than 1, meaning that the original series converges for all p.

Therefore, in interval notation, the series converges for p in interval (-∞, ∞).