Does the series from 0 to infinity of [1/square root of (n+4)] x cos(n x pi) converge or diverge? I think it diverges, but I don't know why.

The "x"s are multiplication signs by the way.

To determine whether the given series converges or diverges, we will use the convergence tests, specifically the nth-term test and the alternating series test.

First, let's examine the nth-term test. According to the nth-term test, if the limit of the nth term of the series as n approaches infinity is not zero, the series diverges.

For our series, the nth term is given by [1/square root of (n+4)] x cos(n x pi). So, we need to find the limit of this term as n goes to infinity.

Let's simplify the term [1/square root of (n+4)] x cos(n x pi):

= cos(n x pi) / square root of (n+4)

Now, we will take the limit of this term as n approaches infinity:

Limit as n approaches infinity of cos(n x pi) / square root of (n+4).

To evaluate this limit, we need to consider the behavior of cos(n x pi) and square root of (n+4) as n goes to infinity.

The cosine function oscillates between -1 and 1 as the angle increases, and does not converge to a single value. Meanwhile, square root of (n+4) approaches infinity as n becomes larger.

Since the numerator (cos(n x pi)) does not have a limit and the denominator (square root of (n+4)) goes to infinity, the limit of the term [cos(n x pi) / square root of (n+4)] as n approaches infinity does not exist. Therefore, the nth term of the series does not approach zero, and the nth-term test fails to provide a definitive answer.

Now, let's utilize the alternating series test. The alternating series test applies when a series alternates between positive and negative terms. In our case, the cosine function guarantees the alternation between positive and negative values for each term of the series.

To satisfy the alternating series test, two conditions must be met:
1. The absolute value of each term must decrease as n increases.
2. The limit of the absolute value of the nth term must approach zero as n approaches infinity.

Looking at the given series, we observe that the absolute value of each term is given by:

| [1/square root of (n+4)] x cos(n x pi) |

Since the cosine function oscillates between -1 and 1, we can ignore the cosine term and focus on the absolute value of [1/square root of (n+4)]. We can express this as:

| 1/square root of (n+4) |

As n increases, the value of (n+4) also increases, meaning the denominator grows larger. Consequently, the absolute value of the term decreases. Since the magnitude of each term decreases as n increases, the first condition of the alternating series test is satisfied.

Now, to determine if the second condition holds, we need to evaluate the limit of the absolute value of the nth term as n approaches infinity:

Limit as n approaches infinity of | [1/square root of (n+4)] x cos(n x pi) |

= Limit as n approaches infinity of | 1/square root of (n+4) |

To evaluate this limit, we consider the behavior of the term 1/square root of (n+4) as n goes to infinity. It is evident that the term approaches zero.

Since the absolute value of the nth term approaches zero as n approaches infinity, the alternating series test confirms that the series converges.

Therefore, the given series converges.