The task involves determining the convergence of two infinite series using the Cauchy root test ($\sqrt[n]{a_n}$-criterion).

The two series are:

1. $\sum_{n=1}^\infty \left( \frac{n-1}{2n+1} \right)^n$
2. $\sum_{n=1}^\infty \left( \frac{n-1}{n+1} \right)^{n(n-1)}$

Let's solve each part step by step.

a) Series $\sum_{n=1}^\infty \left( \frac{n-1}{2n+1} \right)^n$:

Using the Cauchy root test:

$L = \limsup_{n \to \infty} \sqrt[n]{a_n},$ where $a_n = \left( \frac{n-1}{2n+1} \right)^n$.

We compute: $\sqrt[n]{a_n} = \frac{n-1}{2n+1}.$

Now, take the limit as $n \to \infty$: $\lim_{n \to \infty} \frac{n-1}{2n+1} = \lim_{n \to \infty} \frac{n(1 - \frac{1}{n})}{2n(1 + \frac{1}{2n})} = \frac{1}{2}.$

Thus: $L = \frac{1}{2}.$

Since $L < 1$, the series converges by the Cauchy root test.

b) Series $\sum_{n=1}^\infty \left( \frac{n-1}{n+1} \right)^{n(n-1)}$:

Using the Cauchy root test again:

Let $a_n = \left( \frac{n-1}{n+1} \right)^{n(n-1)}$. We compute: $\sqrt[n]{a_n} = \left( \frac{n-1}{n+1} \right)^{n-1}.$

Now take the logarithm to simplify the asymptotics: $\ln \left( \sqrt[n]{a_n} \right) = (n-1) \ln \left( \frac{n-1}{n+1} \right).$

As $n \to \infty$, expand $\ln \left( \frac{n-1}{n+1} \right)$ using a Taylor series: $\ln \left( \frac{n-1}{n+1} \right) = \ln \left( 1 - \frac{2}{n+1} \right) \sim -\frac{2}{n+1}.$

So: $\ln \left( \sqrt[n]{a_n} \right) \sim (n-1) \cdot \left( -\frac{2}{n+1} \right) = -\frac{2(n-1)}{n+1}.$

As $n \to \infty$: [ \ln \left( \sqrt[n]{a_n} \right) \to -2, \quad \text{so } \sqrt[n]{a_n} \to e^{-2}. ]

Thus: $L = e^{-2}.$

Since $L < 1$, the series converges by the Cauchy root test.

Summary of Results:

1. Series (a) converges.
2. Series (b) converges.

Would you like me to elaborate on any step or explain the reasoning further?

Follow-up Questions:

1. What is the difference between the Cauchy root test and other convergence tests (e.g., ratio test)?
2. Can you apply the Cauchy root test to a non-positive series?
3. How would the result change if $L = 1$ in the Cauchy root test?
4. What are the implications of the Taylor expansion approximation used here?
5. How does the growth of $n^n$ compare to exponential terms like $e^n$ in these tests?

Tip:

When using the Cauchy root test, always check if the limit $L$ is strictly less than 1 for guaranteed convergence. If $L = 1$, the test is inconclusive.