Problem 95 - Essential Calculus

Question

Prove that if the nonnegative series \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) converge, then so does the series \(\sum_{n=1}^{\infty} a_{n} b_{n}\).

Step-by-Step Solution

Answer

The product of two convergent nonnegative series is also a convergent series.

Step 1 : Preliminary Analysis

We know that \(a_n\) and \(b_n\) are nonnegative and converge. They must therefore be bounded, i.e., there exists \(M \geq 0\) such that \(0 \leq a_n, b_n \leq M\) for every \(n \geq 1\). Also, because \(a_n\) and \(b_n\) are nonnegative and converge, their series \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) also converge, which means the sequence of their partial sums is bounded.

Step 2: Setting up the proof by contradiction

Assume, on the contrary, that the series \(\sum_{n=1}^{\infty} a_n b_n\) diverges. This means that the sequence of partial sums of the series is unbounded, i.e., there exists a subsequence of partial sums that is strictly increasing and goes to +∞.

Step 3: Finding the contradiction

But each term in the series \(\sum_{n=1}^{\infty} a_n b_n\) is less than or equal to \(M^2\). So the sequence of partial sums of this series is increasing and bounded above by \(N*M^2\), which leads to a contradiction as a bounded sequence can't go to infinity.

Step 4: Conclusion

Therefore, the assumption that the series \(\sum_{n=1}^{\infty} a_n b_n\) diverges is false, and thus, the series \(\sum_{n=1}^{\infty} a_n b_n\) converges.

Key Concepts

Nonnegative Series

In mathematical terms, a **nonnegative series** is a series where all its terms are nonnegative numbers. This means each term of the series is either zero or a positive number. The property of being nonnegative heavily influences the behavior and convergence of the series.

• Since each term is nonnegative, the partial sums in the series can never decrease. They either stay the same or increase.
• Convergence of a nonnegative series can be determined by checking if the sequence of its partial sums has a finite limit as the number of terms goes to infinity.

The series \(\sum_{n=1}^{\infty} a_n\)and \(\sum_{n=1}^{\infty} b_n\)in the problem are both nonnegative series. Each is composed of terms that do not go below zero, and the fact that they converge implies that their partial sums stabilize at some level rather than growing indefinitely.

Partial Sums

The concept of **partial sums** is crucial for understanding series and their convergence. Each partial sum in a series is the sum of the first few terms of that series. For a series \(\sum_{n=1}^{\infty} a_n\), the partial sum \(S_N\) is expressed as:\[S_N = a_1 + a_2 + \cdots + a_N.\]These partial sums help us determine if a series converges by checking whether these sums approach a certain limit as \(N\) increases. If the sequence of partial sums \(\{S_N\}\) approaches a finite number, then the series is said to converge.

• In the scenario where both \(\sum_{n=1}^{\infty} a_n\)and \(\sum_{n=1}^{\infty} b_n\)converge, it suggests their partial sums each approach a specific finite value.

Thus, the convergence of the series \(\sum_{n=1}^{\infty} a_n b_n\)is analyzed by investigating whether its sequence of partial sums remains bounded as \(N\) increases, reinforcing that all terms in the series are restricted by known limitations derived from the terms \(a_n\) and \(b_n\).

Bounded Sequences

A sequence is considered **bounded** if there exists a real number that provides an upper limit to all terms in the sequence. If no term in the sequence exceeds this maximum number, we can understand it as bounded.For instance, if we can find an \(M\) such that \(0 \leq a_n, b_n \leq M\) for all terms \(a_n\) and \(b_n\), then both sequences are bounded.

• In the exercise, knowing \(a_n\) and \(b_n\) are bounded is essential because it implies stability in their sums.
• This bounding allows the characterization of the terms of the series \(a_n b_n\), allowing us to predict their behavior more reliably.

In a sequence of partial sums, if the sequence is increasing and bounded above, this effectively guarantees that it converges to a finite number. This finding arises in the problem when analyzing the series \(\sum_{n=1}^{\infty} a_n b_n\)and realizing that an increasing and bounded sequence cannot diverge, refuting the assumption of divergence.