Problem 16 - A Course in Calculus and Real Analysis

Question

Suppose the partial sums of a series \(\sum_{k=1}^{\infty} b_{k}\) are bounded. If \(p>0\) and \(x \in(0,1)\), then show that the series $$ \sum_{k=1}^{\infty} \frac{b_{k}}{k^{p}}, \quad \sum_{k=1}^{\infty} \frac{b_{k}}{(\ln k)^{p}} \quad \text { and } \quad \sum_{k=1}^{\infty} b_{k} x^{k} $$ are convergent. (Hint: Proposition \(9.20 .\) )

Step-by-Step Solution

Answer

Given that the partial sums of \(\sum_{k=1}^{\infty} b_{k}\) are bounded, we have for any \(\epsilon > 0\), integers \(M\) and \(N\) that ensure the tails of the series are bounded. We use Cauchy's criterion to show convergence for all three series: 1. \(\sum_{k=1}^{\infty} \frac{b_{k}}{k^{p}}\): The convergence is satisfied by bounding the tails with \(\frac{1}{(m+1)^p}\left|\sum_{k=m+1}^{n} b_{k}\right|\) for all \(n>m\ge M'\), where \(M' = \max\{M,N\}\). 2. \(\sum_{k=1}^{\infty} \frac{b_{k}}{(\ln k)^{p}}\): Similar to the first case, the convergence is satisfied by bounding the tails with \(\frac{1}{(\ln(m+1))^p}\left|\sum_{k=m+1}^{n} b_{k}\right|\) for all \(n>m\ge M'\), where \(M' = \max\{M,N\}\). 3. \(\sum_{k=1}^{\infty} b_{k}x^{k}\): The series converges by the ratio test, as \(\lim_{k \to \infty} \left|\frac{b_{k+1}x^{k+1}}{b_{k}x^{k}}\right| < 1\), given \(x \in (0, 1)\) and \(\{b_k\}\) is bounded. Hence, all three series are convergent.

Step 1: 1. Bounded Partial Sums of Original Series

Given that the partial sums of the series \(\sum_{k=1}^{\infty} b_{k}\) are bounded, we know that for any \(\epsilon > 0\), there exists an integer \(N\) such that for all \(n > m \ge N\), we have \(\left|\sum_{k=m+1}^{n}b_k\right| < \epsilon\).

Step 2: 2. Convergence of the first series

To show that first series is convergent, we need to prove that \(\sum_{k=1}^{\infty} \frac{b_{k}}{k^{p}}\) satisfies Cauchy's criterion. So for any \(\epsilon > 0\), consider the tail of the series: $$ \left|\sum_{k=m+1}^{n} \frac{b_{k}}{k^{p}}\right| \le \sum_{k=m+1}^{n} \left|\frac{b_{k}}{k^{p}}\right| \le \frac{1}{(m+1)^p}\left|\sum_{k=m+1}^{n} b_{k}\right|. $$ Since the partial sums of \(\sum b_{k}\) are bounded and using the condition from step 1, for a given \(\epsilon > 0\), we can find integers \(M\) and \(N\) such that when \(m \ge M\), $$ \left|\sum_{k=m+1}^{n} b_{k}\right| < (m+1)^{p}\epsilon. $$ Let \(M' = \max\{M,N\}\). Then for all \(n > m \ge M'\), we have: $$ \left|\sum_{k=m+1}^{n} \frac{b_{k}}{k^{p}}\right| < \epsilon. $$ Thus, the first series converges by Cauchy's criterion.

Step 3: 3. Convergence of the second series

For the second series, we have: $$ \left|\sum_{k=m+1}^{n} \frac{b_{k}}{(\ln k)^{p}}\right| \le \sum_{k=m+1}^{n} \left|\frac{b_{k}}{(\ln k)^{p}}\right| \le \frac{1}{(\ln(m+1))^{p}}\left|\sum_{k=m+1}^{n} b_{k}\right|. $$ As in the previous case, we can find integers \(M\) and \(N\) such that for a given \(\epsilon > 0\) and \(m \ge M\), $$ \left|\sum_{k=m+1}^{n} b_{k}\right| < (\ln(m+1))^{p}\epsilon. $$ Let \(M' = \max\{M,N\}\). Then for all \(n > m \ge M'\), we have: $$ \left|\sum_{k=m+1}^{n} \frac{b_{k}}{(\ln k)^{p}}\right| < \epsilon. $$ Thus, the second series converges by Cauchy's criterion.

Step 4: 4. Convergence of the third series

For the third series, we can apply the ratio test: $$ \lim_{k \to \infty} \left|\frac{b_{k+1}x^{k+1}}{b_{k}x^{k}}\right| = \lim_{k \to \infty}\left|\frac{b_{k+1}}{b_{k}}\right|x = x\lim_{k \to \infty}\left|\frac{b_{k+1}}{b_{k}}\right|. $$ Since \(x \in (0, 1)\) and \(\{b_k\}\) is a bounded sequence, we have: $$ \lim_{k \to \infty} \left|\frac{b_{k+1}x^{k+1}}{b_{k}x^{k}}\right| < 1. $$ Thus, by the ratio test, the third series converges. In conclusion, all three series, \(\sum_{k=1}^{\infty} \frac{b_{k}}{k^{p}}\), \(\sum_{k=1}^{\infty} \frac{b_{k}}{(\ln k)^{p}}\), and \(\sum_{k=1}^{\infty} b_{k}x^{k}\), are convergent.

Key Concepts

Cauchy's Criterion for Series Convergence

Understanding when a series converges is crucial in the realm of mathematics, especially in a calculus or real analysis course. Cauchy's criterion provides a straightforward yet powerful condition to determine the convergence of infinite series.

A series \(\sum_{k=1}^{\text{\infty}} a_k\) satisfies Cauchy's criterion if for every positive number \(\epsilon\), there exists an integer \(N\) such that for all natural numbers \(m\) and \(n\) greater than \(N\), the absolute value of the sum of terms from \(a_{m+1}\) to \(a_n\) is less than \(\epsilon\). Symbolically, we express this as:

\[ \left|\sum_{k=m+1}^{n} a_k \right| < \epsilon \quad \text{whenever } n > m \geq N. \]
This condition implies that the 'tail' of the series becomes arbitrarily small as you add up more terms, signaling convergence. In the step-by-step solution provided, we see this criterion used to successfully demonstrate the convergence of the series \(\sum_{k=1}^{\text{\infty}} \frac{b_{k}}{k^{p}}\) by comparing it to the bounded partial sums of the series \(\sum_{k=1}^{\text{\infty}} b_{k}\).

Understanding Bounded Sequences

A sequence \(\{a_n\}\) is said to be bounded if there exists a real number \(M\) such that \(\left|a_n\right| \leq M\) for all terms in the sequence. This implies that the sequence does not 'blow up' to infinity or 'dive down' to negative infinity; it stays within a certain range.

In the context of the exercise, it is given that \(\sum_{k=1}^{\text{\infty}} b_{k}\) has bounded partial sums. This means that there exists an upper bound on the absolute sum \(\left|\sum_{k=1}^{n} b_{k}\right|\), for all \(n\). As a consequence, one can leverage this property to assure that modifications of the original series involving extra factors still maintain convergence, as shown in the step-by-step solution.

Bounded sequences play a significant role in analysis because they often lead to convergent subsequences, an idea stemming from the Bolzano–Weierstrass theorem. This pivotal concept helps mathematicians infer the behavior of various math constructs, not just series but also functions and integrals.

The Ratio Test for Convergence

The Ratio Test is another key tool in determining whether an infinite series converges. It is particularly useful for series whose terms involve factorials, exponential functions, or any form where terms grow or shrink at a factorial or exponential rate.

The test states that a series \(\sum_{k=1}^{\text{\infty}} a_{k}\) converges absolutely if:

\[ \lim_{k \to \text{\infty}} \left| \frac{a_{k+1}}{a_{k}} \right| = L < 1. \]
The third series in the solution \(\sum_{k=1}^{\text{\infty}} b_{k}x^{k}\) was shown to converge using the Ratio Test. Since \(x\) lies between 0 and 1, and the sequence \(\{b_k\}\) is bounded, the limit of the ratio of successive terms is less than 1. Therefore, according to the Ratio Test, this series converges.