Problem 4 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\ln (n+1)} $$

Step-by-Step Solution

Answer

The given series converges.

Step 1: Identify the series

The series given is an alternating series, denoted as: \[ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\ln (n+1)} \]

Step 2: Check for decreasing values

The series \[ \frac{1}{\ln(n+1)} \] decreases as \( n \) increases from 1 to infinity. This can be shown by examining the derivative of \( \frac{1}{\ln(n+1)} \) which is \( - \frac{1}{(n+1) \ln^2(n+1)} \). As \( n \) increases, this derivative is negative, hence, \(\frac{1}{\ln(n+1)}\) is decreasing.

Step 3: Check for limit

As \( n \) approaches infinity, \( \frac{1}{\ln(n+1)} \) approaches zero. This can be shown by the limit: \[ \lim_{n\to\infty} \frac{1}{\ln(n+1)} = 0 \]

Step 4: Apply the Alternating Series Test

The given series meets the two criteria of the Alternating Series Test: (1) The terms decrease in absolute value, (2) The limit of the terms as \( n \) approaches infinity is zero. Therefore, by the Alternating Series Test, the series \[ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\ln (n+1)} \] converges.

Key Concepts

Alternating Series Test

Understanding the Alternating Series Test (AST) is essential for determining the convergence of series with alternating positive and negative terms. This test is applied when you have a series of the form: \[ \sum_{n=1}^{\infty} (-1)^n a_n \]
Where the sequence of terms \( a_n \) is always positive. There are two main conditions to check for the AST:

• The terms \( a_n \) must be decreasing: \( a_{n+1} \leq a_n \).
• The limit of the terms must approach zero as \( n \) tends to infinity: \[ \lim_{n\to\infty} a_n = 0 \].

If both conditions are met, the series converges. In our exercise, the AST confirms convergence since \( a_n = \frac{1}{\ln(n+1)} \) is both decreasing and approaches zero.

Infinite Series

An infinite series is a sum of infinitely many terms, expressed as: \[ \sum_{n=1}^{\infty} a_n \].
An important aspect of infinite series is the concept of partial sums. A partial sum \( S_N \) is the sum of the first \( N \) terms of the series: \[ S_N = \sum_{n=1}^{N} a_n \].
The behavior of these partial sums as \( N \) increases without bound determines the convergence or divergence of the series. When the sequence of partial sums has a finite limit, we say the series converges to that limit. However, if the limit of the partial sums doesn't exist or is infinite, the series diverges. The alternating series given in the example is a specific type of infinite series where the sign alternates between successive terms.

Limits of Sequences

The limit of a sequence is the value that the sequence's terms approach as the index \( n \) becomes very large. Mathematically, the limit of the sequence \( \{a_n\} \) is \( L \) if for every positive number \( \epsilon \), there exists a positive integer \( N \) such that for all \( n \geq N \), the terms of the sequence are within \( \epsilon \) of \( L \). This is symbolically represented as: \[ \lim_{n\to\infty} a_n = L \].
In the context of the given exercise, we look at the limit: \[ \lim_{n\to\infty} \frac{1}{\ln(n+1)} \]
By finding that this limit is zero, we satisfy one of the conditions for the Alternating Series Test.

Convergence and Divergence of Series

The concepts of convergence and divergence are critical in the analysis of series. A series converges if the sequence of its partial sums \( \{S_N\} \) approaches a finite value as \( N \) increases indefinitely. On the other hand, a series diverges if the sequence of partial sums fails to converge to a finite limit.
Convergent series can be bounded and used in practical applications such as calculating sums to a certain accuracy. Divergent series, while not summing to a finite number, can still be manipulated and studied for their properties under various mathematical contexts. The convergence of the alternating series in the problem is guaranteed by fulfilling the criteria established by the Alternating Series Test.