Q12Mathematics I

When evaluating the convergence of an infinite mathematical series $\sum u_n$, the most standard initial approach is D'Alembert's Ratio Test, which evaluates the limit ${\lim }_{n\to \infty }\left|\frac{u_{n+1}}{u_n}\right|$. If this limit equals exactly 1, the Ratio Test completely fails to provide a conclusion. In such specific boundary cases, more sensitive mathematical tools, such as Raabe's Test, must be deployed.

Step 1: Applying Raabe's Test Formula

Raabe's Test strictly defines the evaluation limit $L$ as:

The strict convergence criteria for Raabe's Test are: - If $L>1$, the series is absolutely convergent. - If $L<1$, the series is absolutely divergent. - If $L=1$, the test fails and even higher-order tests (like logarithmic tests) are required.

Step 2: Evaluating the Given Sequence

For our specific mathematical problem, the inverse ratio of the sequential terms $\frac{u_n}{u_{n+1}}$ has been algebraically determined to be $\frac{2n+2}{2n+1}$. We directly substitute this rational fraction into Raabe's formula:

We must algebraically simplify the complex expression inside the parentheses by finding a common denominator:

The $2n$ terms in the numerator cancel perfectly, and $2-1=1$, leaving:

Step 3: Calculating the Infinite Limit

To evaluate this standard limit at infinity, we rigorously divide both the numerator and the denominator by the highest power of $n$ (which is $n^1$):

As $n$ tends strictly to infinity, the fraction $1/n$ precisely approaches zero. Thus, the limit resolves exactly to a constant value:

Because the exact calculated limit $L=1/2$ is strictly less than $1$, Raabe's Test mathematically concludes that the given infinite series is definitively divergent.