Question

How many of the following series converge? 1. ∑_{n=2}^{[infinity]} (1 / (n^0.9 * ln(n))) 11. ∑_{n=2}^{[infinity]} (1 / (n^1.1 * ln(n))) III. ∑_{n=2}^{[infinity]} (1 / (n * ln(n)))

Answer 1

In summary, two out of the three series expansions converge. The first series diverges, while the second and third series, due to their respect to the p-series rules and logarithmic involvement, are determined to be convergent.

Step-by-step explanation:

To determine the convergence of the series expansions, we can apply the integral test, since each term in the given series is positive, continuous, and decreasing for n >= 2. The integral test can be used for series of the form Σ_{n=a}^{∞} f(n) if the integral ∫ f(x) dx from a to infinity converges.

In the case of these series expansions, we have:

1. Σ_{n=2}^{∞} (1 / (n^0.9 * ln(n)))
2. Σ_{n=2}^{∞} (1 / (n^1.1 * ln(n)))
3. Σ_{n=2}^{∞} (1 / (n * ln(n)))

For I, the exponent of n is less than 1, and this integral will diverge because it is comparable to the p-series where p <= 1. Therefore, this series does not converge.

For II, the exponent of n is greater than 1, which makes the integral and thus the series convergent because it behaves like a p-series with p > 1.

For III, the exponent of n is 1, but since it also has a logarithmic factor in the denominator, the rate of decrease is slower than the harmonic series, and hence it converges more slowly. This requires checking with the integral test explicitly, and it can be shown that this series converges (e.g., by using Comparison Test or Integral Test).

So, out of the three series given, two of them converge: the second and the third.