\(\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n)^{3 / 2}}\)

Series convergence is when the sum of the terms in an infinite series approaches a fixed number as more terms are added. In maths, we check if adding up all terms in a series will result in a finite value. It’s like having an endless list of numbers, but when you add them all up, they don’t just keep getting bigger without bound.

For a series to converge, it must meet certain criteria. One common method is the nth-term test, where if the limit of the nth term as n approaches infinity is not zero, the series diverges. But if it is zero, we need further tests.

In the exercise, the series \(\frac{(-1)^n x^{2n}}{(2n)^{3/2}}\) converges. Both steps in the test are satisfied, meaning its sum approaches a fixed value.

The Leibniz test, also known as the Alternating Series Test, is a method to determine the convergence of alternating series. An alternating series is a series where the terms alternate in sign, like positive, negative, positive, negative, etc. The test states that an alternating series \(\sum (-1)^n b_n\) converges if two conditions are met:

• The absolute value of the terms \((b_n)\) decreases monotonically, i.e., each term is smaller than the previous term in absolute value.
• The limit of the terms \((b_n)\) as n approaches infinity is 0.

The alternating series in our exercise, \(\frac{(-1)^n x^{2n}}{(2n)^{3/2}}\), meets these conditions, confirming its convergence.

The Alternating Series Test helps in analyzing series where the sign alternates between terms. It’s another name for the Leibniz test and is very useful. It particularly focuses on alternating series.

The test checks two critical conditions:

• The series terms decrease in absolute value, termed as being monotonically decreasing. This ensures no term in absolute value is larger than the one before it.
• The limit of the terms \((a_n)\) as n approaches infinity equals zero. Without this, the terms wouldn't collectively settle to a finite sum.

In the given exercise, we distinguished the series \(\frac{(-1)^n x^{2n}}{(2n)^{3/2}}\) and verified both key points. Terms decrease, and the limit converges to zero.

A sequence is an ordered list of numbers or terms. Each number in the list is called a term. Sequences can be finite or infinite, and they often follow a specific rule or pattern.

In mathematical analysis, sequences are important because they form the backbone of series. If you sum up the terms of a sequence, you get a series. For example, the sequence in the exercise is \(\frac{x^{2n}}{(2n)^{3/2}}\). Each term follows this precise formula.

Understanding sequences helps us analyze their behavior as the index n increases to infinity. In convergence tests, we often examine how the terms of these sequences behave to determine if the series will reach a finite sum.