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A power series is a crucial mathematical concept that involves an infinite sum of terms. Each term is usually composed of a coefficient and a variable raised to a power. The general form could be expressed as:

• \( \sum_{n=0}^{\infty} c_n (x-a)^n \)

Here, \( c_n \) is the coefficient for each term, \( (x-a) \) denotes the variable and its power, and \( a \) is the center of the series.
In the given problem, the power series is written as:

• \( 1 + x + \frac{x^2}{\sqrt{2}} + \frac{x^3}{\sqrt{3}} + \cdots \)

This is a power series centered at zero, since it includes terms of the variable \( x \) raised to various powers. Each term becomes increasingly complex, indicating a pattern where the nth term \( a_n = \frac{x^n}{\sqrt{n}} \) follows. Understanding this frame helps in identifying the convergence, which tells us where the series builds up to a finite sum.

The Absolute Ratio Test is a method used to determine the convergence or divergence of a power series. It involves examining the ratio of successive terms in an absolute sense. The test states:

• A series \( \sum a_n \) converges absolutely if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \).

For the given series, we formulate the ratio of consecutive terms as:

• \( \frac{a_{n+1}}{a_n} = \frac{x^{n+1}}{\sqrt{n+1}} \cdot \frac{\sqrt{n}}{x^n} \)

After simplification, this ratio is \( x \cdot \frac{\sqrt{n}}{\sqrt{n+1}} \). The absolute value result becomes \( |x| \cdot \frac{\sqrt{n}}{\sqrt{n+1}} \).
As \( n \) approaches infinity, the fraction \( \frac{\sqrt{n}}{\sqrt{n+1}} \) tends toward 1. Therefore, the limit to consider becomes \( |x| \), which is central to determining when the series converges.

The convergence set of a power series is the range of values for which the series converges. For the series given, we've determined that \( \lim_{n \to \infty} |x| \cdot \frac{\sqrt{n}}{\sqrt{n+1}} = |x| \).
According to the Absolute Ratio Test, convergence happens when the final limit \( |x| < 1 \). This means the series converges absolutely in the interval \((-1, 1)\).
The convergence set is therefore the set of all \( x \) values between \(-1\) and \(1\), excluding the endpoints. This result shows that within this interval, the series will sum to a finite value, providing a useful tool for understanding function behavior near zero. The identification of a convergence set is crucial in analyzing the behavior of series and their applications in mathematical modeling.