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The Root Test is a powerful tool to determine whether a series converges or diverges. It is especially useful when dealing with series raised to the power of the index variable, like our original problem. To apply the Root Test, we first find the k-th root of the general term of the series.

To do this, consider the general term of the series, noted as \(a_k\). The Root Test involves calculating the limit:

• Find \(\lim_{k\to\infty} \sqrt[k]{a_k}\).

The rule is simple:

• If this limit is less than 1, then the series converges.
• If it is greater than 1, then the series diverges.
• If equal to 1, the test is inconclusive.

In our problem, after computing the limit of \(\sqrt[k]{\left(\frac{k}{2k+1}\right)^k}\), we find it reduces to \(\frac{1}{2}\), which is less than 1.
This means by the Root Test, the series converges!

The limit of a sequence is foundational in calculus and analysis. It helps to understand how the terms of a sequence behave as the index goes to infinity.

In our series problem, once we simplified \(\sqrt[k]{\left(\frac{k}{2k+1}\right)^k}\) to \(\frac{k}{2k+1}\), we needed to evaluate \(\lim_{k\to\infty} \frac{k}{2k+1}\).

To find this limit, look at the leading terms (the terms that grow the fastest) in the numerator and the denominator. For large \(k\), \(\frac{k}{2k+1}\) simplifies to \(\frac{1}{2}\), since:

• The coefficient of \(k\) in the numerator (1) and the dominant part of the expression in the denominator is \(2k\).

So, dividing both by \(k\), only dominant terms matter, leaving \(\frac{1}{2}\). This limit tells us how the ratio behaves as \(k\) gets very large, impacting convergence assessment.

The divergence of series is a crucial concept in understanding infinite sums and their behavior. A series diverges if its partial sums do not approach a finite limit as the number of terms increases.

In the context of the given series, we determined whether the series converges using the Root Test. The notion of divergence comes into play if the k-th root limit were greater than 1. In such a case:

• The terms grow larger, and thus, the partial sums do not settle to a particular size and keep increasing.

Remember, a series diverges if:

• The terms themselves do not approach zero, or
• After applying tests like the Root Test, the conditions for convergence are not met.

This is different from the test being inconclusive, as it implies growth without bound. For our specific case, since the limit \(\frac{1}{2}\) is less than 1, the series converges, meaning it does not diverge.