Problem 49 - Calculus: One and Several Variables

Question

Let \(\sum a_{k}\) be is series with nonneyalive terms. (a) Show that if \(\sum a_{k}\) converges, then \(\sum a_{k}^{2}\) convery. (b) Give an example where \(\sum a_{k}^{2}\) converges and \(\sum a_{k}\) converges; give an example where \(\sum a_{k}^{2}\) converges but \(\sum a_{k}\) diverges.

Step-by-Step Solution

Answer

(a) By applying the Cauchy-Schwarz inequality and setting \(b_k = 1\) for all \(k\), we prove that if \(\sum a_k\) converges, then \(\sum a_k^2\) also converges. (b) An example of a series where both \(\sum a_k\) and \(\sum a_k^2\) converge is the geometric series with \(a_k = \frac{1}{2^k}\). An example of a series where \(\sum a_k^2\) converges, but \(\sum a_k\) diverges, is the harmonic series with \(a_k = \frac{1}{k}\).

Step 1: Apply the Cauchy-Schwarz inequality

One helpful theorem we can use is the Cauchy-Schwarz inequality, which states that for any two sequences \(\{a_k\}\) and \(\{b_k\}\), we have \(\left(\sum_{k=1}^n a_k b_k\right)^2 \leq \left(\sum_{k=1}^n a_k^2\right)\left(\sum_{k=1}^n b_k^2\right)\) Set \(b_k = 1\) for all \(k\), and we get \[\left(\sum_{k=1}^n a_k\right)^2 \leq \left(\sum_{k=1}^n a_k^2\right)\left(\sum_{k=1}^n 1^2\right).\] Now, let the partial sums of \(\sum a_k\) be \(S_m = \sum_{k=1}^m a_k\). It converges, so for each \(\epsilon > 0\), there exists an integer \(M_\epsilon > 0\) such that for all \(m, n > M_\epsilon\), we have \(|S_m - S_n| < \epsilon\).

Step 2: Prove the convergence of \(\sum a_k^2\)

We want to show that \(\sum a_k^2\) converges. In order to prove this, we need to show that for each \(\epsilon > 0\), there exists \(N_\epsilon > 0\) such that for all \(m, n > N_\epsilon\), we have \(\left|\sum_{k=m+1}^{n} a_k^2\right| < \epsilon\). Since \(a_k \geq 0\), we have \(\sum_{k=m+1}^{n} a_k^2 \leq \sum_{k=m+1}^{n} a_k \cdot a_k \leq \sum_{k=m+1}^{n} a_k \cdot S_{M_\epsilon}\), Applying the inequality we derived above, we get \(\sum_{k=m+1}^{n} a_k^2 \leq S_{M_\epsilon}\left|\sum_{k=m+1}^{n} a_k\right| = S_{M_\epsilon}|S_n - S_m|\). Now, since \(S_n\) and \(S_m\) are bounded by \(\epsilon\) whenever \(m, n > M_\epsilon\), we have \(\sum_{k=m+1}^{n} a_k^2 \leq S_{M_\epsilon} \epsilon\). Thus, we can set \(N_\epsilon = M_\epsilon\), and this proves that \(\sum a_k^2\) converges. #Part (b): Examples of different convergence cases#

Step 3: Convergent series for both \(\sum a_k\) and \(\sum a_k^2\)

We need to find an example of a series where both \(\sum a_k\) and \(\sum a_k^2\) converge. Consider the geometric series with \(a_k = \frac{1}{2^k}\). We have \(\sum_{k=1}^\infty \frac{1}{2^k} = 1\), and \(\sum_{k=1}^\infty \left(\frac{1}{2^k}\right)^2 = \sum_{k=1}^\infty \frac{1}{4^k} = \frac{1}{3}\). Both series converge, so we have our example in this case.

Step 4: Convergent series for \(\sum a_k^2\), but divergent for \(\sum a_k\)

We need to find an example of a series where \(\sum a_k^2\) converges, but \(\sum a_k\) diverges. Consider the harmonic series with \(a_k = \frac{1}{k}\). We know that the harmonic series diverges, i.e., \(\sum_{k=1}^\infty \frac{1}{k}\) diverges. However, the series with squared terms converges because it is a p-series with \(p = 2 > 1\): \(\sum_{k=1}^\infty \left(\frac{1}{k}\right)^2 = \sum_{k=1}^\infty \frac{1}{k^2} < \infty\). Thus, we have our example in this case.

Key Concepts

Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality is a powerful mathematical tool used in many areas, including series convergence problems. It provides a useful way to compare and bound sums of sequences. The inequality states that for any two sequences \( \{a_k\} \) and \( \{b_k\} \), the following holds:
\[ \left(\sum_{k=1}^n a_k b_k\right)^2 \leq \left(\sum_{k=1}^n a_k^2\right)\left(\sum_{k=1}^n b_k^2\right) \]
In the context of series convergence, we can often set \(b_k = 1\) to simplify calculations. By choosing \(b_k = 1\), the inequality becomes:
\[ \left(\sum_{k=1}^n a_k\right)^2 \leq n \sum_{k=1}^n a_k^2 \]
This adjusted inequality helps demonstrate the convergence behavior of the series \(\sum a_k^2\) when \(\sum a_k\) converges. The important takeaway is that the inequality provides us a way to compare the behavior of these series, ensuring we can establish bounds that prove convergence.

Geometric Series

A geometric series is a specific type of infinite series where each term is a constant multiple of the previous one. This series takes the form:
\[ \sum_{k=0}^\infty ar^k = a + ar + ar^2 + ar^3 + \ldots \]
where \(a\) is the first term and \(r\) is the common ratio. The convergence of a geometric series depends on the value of the common ratio \(r\):

• If \(|r| < 1\), the series converges to \( \frac{a}{1-r} \).
• If \(|r| \geq 1\), the series diverges.

The geometric series is an important concept in calculus and analysis because it is one of the simplest types of series that demonstrate convergence, depending on the common ratio \(r\). As shown in the problem solution, a series like \(\sum \frac{1}{2^k}\) converges because its ratio is less than 1.

Harmonic Series

The harmonic series is one of the most famous examples of a divergent series. It is defined as:
\[ \sum_{k=1}^\infty \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots \]
Despite the terms getting smaller, the series does not converge to a finite limit. This means the series infinitely grows without bound.
Understanding the harmonic series is crucial because it provides insight into the behavior of series with decreasing terms. The fact that it diverges while \( \sum \frac{1}{k^2} \), a p-series with \(p = 2\), converges shows that small adjustments in the power of terms can dramatically affect convergence.

p-Series

A p-series is a type of series that can be written as:
\[ \sum_{k=1}^\infty \frac{1}{k^p} \]
where \(p\) is a positive constant. The convergence of a p-series depends critically on the value of \(p\):

• If \(p > 1\), the series converges.
• If \(p \leq 1\), the series diverges.

This convergence criterion makes p-series a fundamental example in the study of series convergence. For instance, the series \(\sum \frac{1}{k^2}\) is convergent because it is a p-series with \(p = 2\), which is greater than 1, showing that the series sums to a finite value. Understanding p-series is essential for grasping how series behave and converge in mathematical analysis.