Question

An infinite product \(P=a_{1} a_{2} a_{3} \ldots,\) which is denoted \(\prod_{k=1}^{\infty} a_{k}\) is the limit of the sequence of partial products \(\left\\{a_{1}, a_{1} a_{2}, a_{1} a_{2} a_{3}, \ldots\right\\} .\) Assume that \(a_{k}>0\) for all \(k\) a. Show that the infinite product converges (which means its sequence of partial products converges) provided the series \(\sum_{k=1}^{\infty} \ln a_{k}\) converges. b. Consider the infinite product $$P=\prod_{k=2}^{\infty}\left(1-\frac{1}{k^{2}}\right)=\frac{3}{4} \cdot \frac{8}{9} \cdot \frac{15}{16} \cdot \frac{24}{25} \cdots$$ Write out the first few terms of the sequence of partial products, $$P_{n}=\prod_{k=2}^{n}\left(1-\frac{1}{k^{2}}\right)$$ (for example, \(P_{2}=\frac{3}{4}, P_{3}=\frac{2}{3}\) ). Write out enough terms to determine the value of \(P=\lim _{n \rightarrow \infty} P_{n}\) c. Use the results of parts (a) and (b) to evaluate the series $$\sum_{k=2}^{\infty} \ln \left(1-\frac{1}{k^{2}}\right)$$

Short answer

#Short answer# a. If the series of natural logarithms of each term converges, then the infinite product converges. b. The first few terms of the given infinite product are: $$P_{2} = \frac{3}{4}\), \(P_{3} = \frac{2}{3}, \(P_{4} = \frac{5}{8}, \(P_{5} = \frac{1}{2}$$. The value of the limit of the sequence of partial products is \(\frac{1}{2}\). c. The sum of the series of natural logarithms of the terms is: $$\sum_{k=2}^{\infty} \ln \left(1-\frac{1}{k^{2}}\right) = -\ln 2$$.

Step-by-step solution

Understand the relationship between the infinite product and its sequence of partial products

We have the infinite product \(\prod_{k=1}^{\infty} a_{k}\). Since the given infinite product is the limit of the sequence of partial products, we need to find the relationship between the convergence of the infinite product and the convergence of the series of logarithms of each term.

Relate the infinite product and its sequence of partial products to the series of logarithms

We can take the natural logarithm of each term in the infinite product and sum them up: $$\sum_{k=1}^{\infty} \ln a_k = \ln(a_1) + \ln(a_2) + \ln(a_3) + \cdots$$. The given condition states that \(\sum_{k=1}^{\infty} \ln a_k\) converges.

Apply properties of logarithms and limits to derive the convergence of the infinite product

Since the series of logarithms converges, we can use properties of logarithms to rewrite the series: $$\ln(a_1) + \ln(a_2) + \ln(a_3) + \cdots = \ln(a_1 a_2 a_3...)$$. By applying the definition of the limit, if \(\sum_{k=1}^{\infty} \ln a_k\) converges to a finite limit L, ie., \(L=\lim_{n \to \infty}\sum_{k=1}^{n} \ln a_k\), then it must also be true that $$\lim_{n \to \infty} \ln(\prod_{k=1}^{n} a_k) = L$$ and thus $$\lim_{n \to \infty} \prod_{k=1}^{n} a_k = e^L$$. Therefore, the infinite product converges. #Phase 2#: b. Find the first few terms of the infinite product and determine the value of its limit

Write out the first few terms of the sequence of partial products

The sequence of partial products for the given infinite product is: $$P_{n} = \prod_{k=2}^{n}\left(1-\frac{1}{k^{2}}\right)$$. Let's calculate the first few terms of the sequence: \(P_{2} = \frac{3}{4}\), \(P_{3} = \frac{3}{4} \cdot \frac{8}{9} = \frac{2}{3}\), \(P_{4} = \frac{2}{3} \cdot \frac{15}{16} = \frac{5}{8}\), \(P_{5} = \frac{5}{8} \cdot \frac{24}{25} = \frac{1}{2}\).

Determine the value of the limit of the sequence of partial products

From the first few terms of the sequence of partial products, we can see that the terms get closer and closer to the value of \(\frac{1}{2}\). Therefore, the value of the limit of the sequence of partial products is: $$P=\lim_{n \to \infty}P_{n}=\frac{1}{2}$$. c. Evaluate the series of natural logarithms of the terms:

Write out the series in terms of the logarithm

We want to evaluate the series: $$\sum_{k=2}^{\infty} \ln \left(1-\frac{1}{k^{2}}\right)$$. Using the result from part (a), the convergence of the series implies the convergence of the infinite product, and we have determined the value of the infinite product to be \(\frac{1}{2}\).

Apply properties of logarithms and limits to find the sum of the series

Since the infinite product converges to a finite value, we can relate the sum of the series to the limit of the sequence of partial products: $$\sum_{k=2}^{\infty} \ln \left(1-\frac{1}{k^{2}}\right) = \lim_{n \to \infty} \ln P_{n}= \ln P$$. We know that \(P = \frac{1}{2}\), so $$\sum_{k=2}^{\infty} \ln \left(1-\frac{1}{k^{2}}\right) = \ln \left(\frac{1}{2}\right) = -\ln 2$$.

Key concepts

Partial Products

When dealing with infinite products, we often talk about partial products, which are like stepping stones to understanding the full product. Imagine you have a sequence of numbers, each greater than zero. The partial product is simply the product of the first few numbers in this sequence. For example, if you have a sequence \( (a_1, a_2, a_3, \ldots) \), then the partial products would be \( a_1, a_1a_2, a_1a_2a_3, \) and so on.

• Partial products help us examine how an infinite product behaves by looking at finite sections of it.
• They are crucial in determining if an infinite product converges, that is, if it approaches a specific value as you multiply more terms.
• By focusing on partial products, mathematicians can study the progression and delicate balancing act of the infinite process.

To summarize, a partial product can be thought of as a way to peep into the whole by just looking at parts, making the concept of infinity more manageable.

Convergence of Series

Convergence of a series is a fascinating concept, especially when linked to infinite products. Think of it as understanding whether the sum of an endless list of numbers reaches a certain limit. For our infinite product to converge, the related series must also converge. This means:

• The series formed by taking the natural logarithm of each term in the product must add up to a finite number.
• Convergence tells us whether the long-term behavior of an infinite sum leads to a stable, specific outcome.

For instance, if the series \( \sum_{k=1}^{\infty} \ln a_k \) converges to a finite limit \( L \), it signifies that the infinite product \( \prod_{k=1}^{\infty} a_k \) will also converge to a value like \( e^L \). Understanding convergence is like ensuring your journey has a destination rather than going forever without settling down.

Natural Logarithm

The natural logarithm, often written as \( \ln \), plays a significant role in connecting series and infinite products. It's a special mathematical function often used for simplifying multiplication into addition by transforming products into sums. Why is it handy here?

• Taking the \( \ln \) of each element in an infinite product transforms our original product into a sum. This shift makes mathematical analysis much simpler.
• It allows us to apply properties of logarithms, such as \( \ln(ab) = \ln(a) + \ln(b) \), which is incredibly useful for breaking complex products into manageable parts.
• The convergence of the series \( \sum_{k=1}^{\infty} \ln a_k \) directly informs us about the convergence of the infinite product.

Thus, the natural logarithm isn't just a calculator button, but a powerful tool that provides insight and understanding into intricate infinite structures.

Limit of Sequence

A limit of a sequence, much like a pause to take stock of a continuously moving train, helps us determine what happens to the sequence as it continues indefinitely. In the world of infinite products, identifying this limit is crucial for understanding convergence.

• When we talk about the limit of a sequence of partial products \( \{P_n\} \), we mean the value that \( P_n \) approaches as \( n \to \infty \).
• If this limit exists and is finite, we then say that the infinite product converges.
• In practice, calculating a few initial terms gives us an idea about where \( P_n \) is heading, making prediction more straightforward.

In our example, by calculating the first few terms \( P_2 = \frac{3}{4}, P_3 = \frac{2}{3}, P_4 = \frac{5}{8}, \) and so on, we observed that the sequence's limit approaches \( \frac{1}{2} \). Understanding limits provides clarity on infinite sequences without having to calculate endlessly and guides us to the beautiful completion of an infinite journey.