The Convergence of the series ((log n)/(log(n+1)))^(n^2 log_n)

Introduction:
In this question, we need to determine whether the series

((log n)/(log(n+1)))^(n^2 log_n)

converges or diverges. To analyze the convergence of the series, we will use various tests and techniques.

Ratio Test:
The ratio test is a common test used to determine the convergence or divergence of series. Let's apply the ratio test to this series:

lim(n→∞) |(a_(n+1))/(a_n)|
= lim(n→∞) |(log(n+1)/(log((n+1)+1)))^((n+1)^2 log(n+1))/(log n)^(n^2 log n)|

Properties of Logarithms:
To simplify the expression, let's use some properties of logarithms.

1. log(a/b) = log(a) - log(b)
2. log(a^b) = b log(a)

Using these properties, we can rewrite the expression as:

lim(n→∞) |(log(n+1) - log(n+2))/(log n)|^((n+1)^2 log(n+1))/(n^2 log n)

Simplifying the Expression:
Now, let's simplify the expression further by expanding the terms:

lim(n→∞) |log(n+1)/(log n) - log(n+2)/(log n)|^((n+1)^2 log(n+1))/(n^2 log n)

Since the logarithm of any number greater than 1 is positive, we can remove the absolute value:

lim(n→∞) (log(n+1)/(log n) - log(n+2)/(log n))^((n+1)^2 log(n+1))/(n^2 log n)

Using Limit Laws:
Now, we can use some limit laws to simplify the expression further. Let's apply the limit laws to both terms separately.

lim(n→∞) log(n+1)/(log n) = 1

lim(n→∞) log(n+2)/(log n) = 1

Therefore, the expression becomes:

lim(n→∞) (1 - 1)^((n+1)^2 log(n+1))/(n^2 log n)

Final Simplification:
Since the limit of (1 - 1) is 0, the expression simplifies to:

lim(n→∞) 0^((n+1)^2 log(n+1))/(n^2 log n) = 0

Conclusion:
The limit of the ratio of consecutive terms of the series is 0. According to the ratio test, if the limit is less than 1, the series converges. In this case, the limit is 0, which is less than 1. Therefore, the given series converges.