\(\sum_{n=1}^{\infty} \frac{x^{2 n}}{2^{n} n^{2}}\)

The ratio test is a common method used to determine the convergence or divergence of an infinite series. It specifically examines the behavior of the ratio of consecutive terms in the series. To apply the ratio test:

• First, recognize the general term of the series, denoted as \( a_n \).
• Next, find the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \)
• Finally, take the limit as \( n \) approaches infinity: \( \lim_{{n\to\to\to\to\to\to\to\to}} \left| \frac{a_{n+1}}{a_{n}} \right| \).

If the limit is less than 1, the series converges absolutely. If the limit is greater than 1 or approaches infinity, the series diverges. If the limit equals 1, the test is inconclusive. In our problem, the general term of the series is \( \frac{x^{2n}}{2^n n^2} \). We calculated \( \left| \frac{a_{n+1}}{a_n} \right| \) and ended up with \( \lim_{{n\to\to\to\to}} \left| \frac{x^{2}}{2}\right| = \frac{x^{2}}{2} \).

The convergence criteria for a series help us to determine under what conditions a series sums to a finite value. Using the ratio test, we've established the following convergence criteria:

• If the ratio \( \lim_{{n\to\infty}} \left| \frac{a_{n+1}}{a_{n}} \right| \) is less than 1, the series converges absolutely.
• If the ratio is greater than 1 or equal to 1, the series diverges or the test is inconclusive.

For \( \sum_{n=1}^{\infty} \frac{x^{2 n}}{2^{n} n^{2}} \), after applying the ratio test, we identified the condition \( \frac{x^{2}}{2} < 1 \). Solving this inequality, we found that \( |x| < \sqrt{2} \). Hence, the series converges when \( |x| < \sqrt{2} \).

In the context of infinite series, limit evaluation plays a vital role, particularly in the ratio test. The limit is used to compare the ratio of successive terms as \( n \) approaches infinity. Consider this problem: initially, we computed \( \frac{a_{n+1}}{a_{n}} \), resulting in \( \frac{x^{2}}{2}\cdot\left( \frac{n}{n+1}\right)^2 \). We then evaluated the limit:
\[ \lim_{{n\to\infty}} \left| \frac{x^{2}}{2}\cdot\left( \frac{n}{n+1}\right)^2\right| = \frac{x^{2}}{2} \cdot \lim_{{n\to\infty}} \left( \frac{n}{n+1} \right)^2 = \frac{x^{2}}{2} \cdot 1 = \frac{x^{2}}{2}\].
Evaluating this limit helped determine the conditions under which the series converges. This technique emphasizes how limits are crucial for defining convergence or divergence in infinite series.

Inequalities are critical in defining the conditions for the convergence of a series. Here, solving the inequality \( \frac{x^{2}}{2} < 1 \) was necessary to interpret the ratio test results. This was done by isolating \(x\):

• Multiply both sides of \( \frac{x^{2}}{2} < 1 \) by 2 to get \( x^2 < 2 \).
• Take the square root of both sides to derive \( |x| < \sqrt{2} \).

This inequality signifies that the series converges absolutely when \( |x| < \sqrt{2} \). Understanding how to handle inequalities helps one derive meaningful results about series behavior, ensuring students grasp the vital steps needed for their problem-solving toolkit.

An infinite series is a sum of an infinite number of terms. In this context, analyzing whether such a sum converges (has a finite value) or diverges (grows indefinitely) is essential. Consider our example: \( \sum_{n=1}^{\infty} \frac{x^{2 n}}{2^{n} n^{2}} \). We're examining the terms from \( n = 1 \) to infinity.

To determine if this infinite series converges, we applied the ratio test. The results, along with resolving relevant inequalities, provided us with the convergence condition \( |x| < \sqrt{2} \).

Understanding infinite series is fundamental in math, especially in calculus and analysis. These concepts frequently appear in various fields, making it vital to grasp their behavior and convergence conditions.