Question

Prove: If \(\sum a_{n}^{2}<\infty\) and \(\sum b_{n}^{2}<\infty,\) then \(\sum a_{n} b_{n}\) converges absolutely.

Short answer

Question: Prove that if two series \(\sum a_n^2 < \infty\) and \(\sum b_n^2 < \infty\) converge, then the series \(\sum a_nb_n\) converges absolutely. Answer: By applying the Cauchy-Schwarz inequality, we show that the sum of the partial sums of the series \(\sum |a_nb_n|\) is bounded, which implies that the series \(\sum |a_nb_n|\) converges, proving that the series \(\sum a_nb_n\) converges absolutely.

Step-by-step solution

Apply the Cauchy-Schwarz inequality

Using the Cauchy-Schwarz inequality for the series \(\sum a_nb_n\), we have: \[(\sum_{n=1}^N a_nb_n)^2 \le (\sum_{n=1}^N a_n^2)(\sum_{n=1}^N b_n^2)\]

Use the definitions of convergence

Since \(\sum a_n^2 < \infty\) and \(\sum b_n^2 < \infty\), there exist some real numbers M and L such that \(\sum_{n=1}^N a_n^2 \le M\) and \(\sum_{n=1}^N b_n^2 \le L\) for all positive integers N. Plugging these into the inequality we derived in Step 1, we get: \[(\sum_{n=1}^N a_nb_n)^2 \le ML\]

Extract the sum of the product

Taking the square root of both sides of the inequality, we obtain: \[\sum_{n=1}^N a_nb_n \le \sqrt{ML}\]

Show that the original series converges absolutely

Now, note that the partial sums of the series \(\sum |a_nb_n|\) are bounded by \(\sqrt{ML}\). In other words, the sum of the absolute values of the terms is bounded, which implies the convergence of the series. Therefore, \(\sum |a_nb_n| < \infty\), and we have proven that the series \(\sum a_nb_n\) converges absolutely.

Key concepts

Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality is a powerful tool in mathematical analysis that provides a way to relate the sum of the product of two sequences to the product of their individual sums of squares.

To understand this inequality, consider two sequences of real numbers, \(a_n\) and \(b_n\). The Cauchy-Schwarz inequality states that for all sequences \(a_n\) and \(b_n\) and for any positive integer \(N\), the following relationship holds:\[\left(\sum_{n=1}^N a_n b_n\right)^2 \leq \left(\sum_{n=1}^N a_n^2\right)\left(\sum_{n=1}^N b_n^2\right)\]This bound becomes extremely useful in real analysis, especially when examining the convergence of series. It effectively allows us to estimate the sum of products \(a_n b_n\) in terms of the sums of squares of \(a_n\) and \(b_n\) separately, which are often easier to evaluate.

It’s important to discern that this inequality does not claim equality but rather provides an upper bound that must be respected. This upper limit paves the path for further analysis in the convergence of series, especially when tested for absolute convergence.

In the case of our textbook problem, the Cauchy-Schwarz inequality is initially applied to set an upper bound for the series of products \(a_n b_n\), bringing us closer to proving absolute convergence.

Series Convergence

In real analysis, understanding series convergence is indispensable. When we say that a series \( \sum a_n\) converges, we imply that the sequence of its partial sums \(S_N = \sum_{n=1}^N a_n\) approaches a finite limit as \(N\) goes to infinity.

To illustrate, if we have a sequence \(a_n\), the series formed by adding its terms, \(\sum a_n\), will either converge or diverge. Convergence means that as you sum more and more terms, the sum will tend towards a specific number, rather than growing without bound or oscillating indefinitely.

Mathematically, for series convergence, we require that for any given small positive number \(\epsilon\), there is a large number \(N\) such that all further partial sums \(S_M\) where \(M > N\) are within \(\epsilon\) of the series' limit. In the context of our exercise, we are looking at the product series \(\sum a_nb_n\) and using the properties of convergence to establish the behavior of this series.

Absolute Convergence

Absolute convergence is a stronger form of convergence for series. A series \(\sum a_n\) is said to converge absolutely if the series of absolute values \(\sum |a_n|\) also converges. Importantly, if a series converges absolutely, it converges normally as well, but the reverse is not always true.

Understanding absolute convergence is critical because it allows us to manipulate series more freely, including reordering terms without changing the series' value. This property stems from the series being well-behaved in the sense that the sum of the absolute values of all terms is finite.

Applying this to our exercise, after using the Cauchy-Schwarz inequality, we show that the sequence of partial sums of \(\sum |a_nb_n|\) is bounded. This indicates that the total sum of the absolute values does not exceed a certain value, providing proof of absolute convergence. Hence, the original series \(\sum a_nb_n\) not only converges but does so absolutely, guaranteeing a certain level of 'predictability' and robustness in its behavior, which is a vital aspect of real analysis.