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A power series is a type of infinite series that takes the form:

• \( \sum_{n=0}^{\infty} a_n x^n \)

It is defined by a sequence \( a_n \) and a variable \( x \). Each term is composed of a coefficient \( a_n \) and a power of \( x \). This kind of series is very versatile and forms the basis of many mathematical analyses, including functions and approximations.
The given series is:

• \( \sum_{n=0}^{\infty} (2^n + n^2) x^n \)

Here, the sequence \( a_n \) is \( 2^n + n^2 \). This represents the coefficients for each term in the power series. Understanding the behavior of these coefficients as the series progresses is crucial in determining convergence properties.

Power series allow the representation of a wide variety of functions once the range (radius) of the variable \( x \) for which the series converges is known.

The ratio test is a popular tool for determining the convergence of infinite series. It works by examining the ratio of successive terms in a series, which can give insight into the behavior of the series as it progresses. For any series \( \sum a_n \), the ratio test involves evaluating:

• \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)

The steps are straightforward:

• Compute the ratio \( \frac{a_{n+1}}{a_n} \).
• Simplify the expression and take the limit as \( n \to \infty \).

Depending on the value of \( L \), the series may converge or diverge:

• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

For the provided power series:

• The ratio \( \frac{a_{n+1}}{a_n} \) was simplified to \( 2 \).

Hence, the radius of convergence can be determined directly from this limit.

Convergence refers to whether or not the sum of an infinite series approaches a finite number. Understanding convergence is crucial in determining if a series like our power series effectively represents a function over a certain interval. By using the ratio test, we calculated the limit \( L = 2 \), which allowed us to determine that:

• If \( x \) is within the radius of convergence, the series converges.

The radius of convergence \( R \) is given by \( \frac{1}{L} \). For our series:

• \( R = \frac{1}{2} \)

This means that the series converges for all \( x \) such that \( |x| < \frac{1}{2} \). The concept of radius of convergence helps in identifying the interval of \( x \) over which the power series will represent a function legitimately. Understanding these intervals allows mathematicians and scientists to reliably use power series in real-world applications.