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The series provided is:

\[
\sum_{n=0}^{\infty} \frac{n+1}{2^n \cdot \left( n^2 + 1 \right)} x^n
\]

To find the **radius of convergence** \(R\), we will use the **Ratio Test**. For a general power series of the form:

\[
\sum_{n=0}^{\infty} a_n x^n
\]

The ratio test states that the radius of convergence is given by:

\[
\frac{1}{R} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
\]

Here, the general term \( a_n = \frac{n+1}{2^n \cdot (n^2 + 1)} \). Let’s apply the ratio test to this expression.

I'll now compute the limit step by step.The limit of the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \) as \( n \to \infty \) is \( \frac{1}{2} \). 

According to the ratio test, the reciprocal of this value gives the radius of convergence \( R \). Hence, the radius of convergence is:

\[
R = 2
\]

Would you like further details or have any questions? Here are some related questions you may find interesting:

1. How does the ratio test work for determining convergence?
2. Can the ratio test be applied to any power series?
3. What happens if the limit of the ratio test equals 1?
4. Are there other methods to find the radius of convergence?
5. What does the radius of convergence tell us about the behavior of the series?

**Tip:** For series with factorials or exponential growth terms, the ratio test is especially useful in determining the radius of convergence.

This problem demonstrates how to find the radius of convergence of the power series sum(n=0 to infinity) (n+1) / (2^n * (n^2 + 1)) * x^n using the Ratio Test. The solution includes key mathematical concepts like power series, convergence, and the ratio test theorem, providing a detailed step-by-step explanation. Suitable for university-level students or advanced high-school learners.

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