Question

Let \(b_{3 m+1}, b_{3 m+2}=-2,\) and \(b_{3 m+3}=1\) for \(m \geq 0\). Show that $$ \sum_{n=1}^{\infty} \frac{b_{n}}{n}=\frac{2}{3} \sum_{m=0}^{\infty} \frac{1}{(m+1)(3 m+1)(3 m+2)} $$

Short answer

Question: Prove the following identity: $$ \sum_{n=1}^{\infty} \frac{b_{n}}{n} = \frac{2}{3} \sum_{m=0}^{\infty} \frac{1}{(m+1)(3 m+1)(3 m+2)} $$ where \(b_{3m+1} =1\), \(b_{3m+2} = -2\), and \(b_{3m+3} =1\). Answer: The identity holds true.

Step-by-step solution

Break down the LHS sum based on the given values of \(b_n\)

We are given that \(b_{3m+1}=1\), \(b_{3m+2}=-2\), and \(b_{3m+3}=1\). We can separate the sum as follows: $$ \sum_{n=1}^{\infty} \frac{b_n}{n} =\sum_{m=0}^{\infty} \left(\frac{1}{3m+1} - \frac{2}{3m+2} + \frac{1}{3m+3}\right) $$

Merge sums and our goal RHS

Our goal is to rewrite the LHS sum into a form where it can be easily compared to the RHS sum: $$ \frac{2}{3} \sum_{m=0}^{\infty} \frac{1}{(m+1)(3 m+1)(3 m+2)} $$ First, let's combine the terms inside the sum on the LHS, under a common denominator: $$ \sum_{m=0}^{\infty} \frac{-3}{(3m+1)(3m+2)}+\frac{2}{(3m+1)(3m+3)}+\frac{1}{(3m+2)(3m+3)} $$

Factor out \(\frac{1}{3}\) from the LHS sum

We can think of factoring out \(\frac{1}{3}\) from the LHS sum, in order to compare it to the RHS sum more easily. Let's do that: $$ \sum_{m=0}^{\infty} \frac{-1}{(3m+1)(3m+2)}+ \frac{2}{(3m+1)(3m+3)} + \frac{1}{(3m+2)(3m+3)} $$ Now multiply and divide by 2 and rewrite the sum: $$ \frac{2}{3} \sum_{m=0}^{\infty} \frac{-1}{2(3m+1)(3m+2)} + \frac{1}{(3m+1)(3m+3)}+ \frac{1}{2(3m+2)(3m+3)} $$

Combine the terms based on the sum's index

To combine the terms in the above sum, add the terms with the same index value (i.e., the same value for \(m\)): $$ \frac{2}{3} \sum_{m=0}^{\infty} \frac{-1 + 2(3m+2) + (3m+2)}{2(3m+1)(3m+2)(3m+3)} $$

Simplify and compare to the RHS

Simplify the numerator of the expression inside the sum: $$ \frac{2}{3} \sum_{m=0}^{\infty} \frac{1}{(m+1)(3m+1)(3m+2)} $$ Now, we can see that the LHS sum has the same format as the RHS sum, which proves the given identity: $$ \sum_{n=1}^{\infty} \frac{b_{n}}{n} = \frac{2}{3} \sum_{m=0}^{\infty} \frac{1}{(m+1)(3m+1)(3m+2)} $$

Key concepts

Infinite Series

An infinite series is essentially the sum of the terms of an infinite sequence. To understand this, imagine adding up an endless list of numbers, one after the other. This may sound like an impossible task, but mathematicians have developed ways to make sense of this process through careful definitions and rules.

In the context of the given exercise, we're dealing with the infinite series \[ \[\begin{equation} \biggl(\frac{b_n}{n}\biggr)_{n=1}^{\infty} \tag{1} \end{equation}\]\] where the terms depend on the values of the sequence \(\{b_n\}\). It's important to note that not all infinite series have a finite sum, which leads us to discuss the concept of convergence.

Summation

The concept of summation refers to the process of adding up elements. The symbol \(\sum\) represents summation in mathematics. It tells us to sum over a range of terms. For finite sums, this is straightforward, but for infinite sums, like the ones in our exercise, we need the concept of limits and convergence to determine what, if anything, the sum approaches as the number of terms tends toward infinity.

In our example, the exercise begins by breaking down the sum based on the sequence's defined values. This stepwise division and manipulation of the series reveals the underlying structure and behavior of the sequence, enabling us to transform the series into a more manageable form that can be analyzed for convergence.

Convergence

In the realm of mathematics, convergence is a cornerstone concept when dealing with infinite series. A series is said to converge if the sum of its terms approaches a specific value as the number of terms grows indefinitely. Conversely, if the sum does not approach any particular value, the series is said to diverge.

The convergence of the series in our exercise is not immediately clear and requires manipulation to reveal. The presented steps systematically transform the left-hand side (LHS) of the given equation into a form that can be readily compared to the right-hand side (RHS). Through this, we aim to show the convergence to the same value on both sides, affirming the exercise's statement.

Sequences

A sequence is a list of numbers in a specific order. In sequences, each number is called a term. In our exercise, the sequence \(\{b_n\}\) is defined by a repeating pattern depending on the index \(n\). Sequences can be finite or infinite, and it is the infinite kind that is most intriguing, as it can lead to an infinite series when we look at the sum of the terms.

Understanding sequences is crucial to solving problems involving series because the behavior of a sequence—whether it's increasing, decreasing, bounded, or follows a specific pattern—can greatly affect the convergence and value of the series it produces.