Question

Suppose that $a_n>0$ for all $n$ and that $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges. Suppose that $b_n$ is an arbitrary sequence of zeros and ones. Does $\sum _{n=1}^{\mathrm{\infty }}{a}_n{b}_n$ necessarily converge?

Short answer

Yes, the series converges regardless of the zeros and ones in $b_n$.

Step-by-step solution

Understand the Given Series

We are given that $a_n>0$ for all $n$ and that the series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges. This means the terms $a_n$ tend to zero as $n$ approaches infinity and the sum of these terms does not diverge.

Introduce Arbitrary Sequence

Consider a sequence $b_n$, where each $b_n$ is either 0 or 1. The series $\sum _{n=1}^{\mathrm{\infty }}{a}_n{b}_n$ is composed of terms $a_n$ from the original series, each multiplied by $b_n$. This modifies the series depending on the values of $b_n$.

Explore Possible Scenarios

Evaluate the impact of $b_n$ on the series. If $b_n=0$ for all $n$, then $\sum _{n=1}^{\mathrm{\infty }}{a}_n{b}_n=0$, clearly converging. However, when $b_n=1$ for infinitely many $n$, we must assess convergence of $\sum _{n=1}^{\mathrm{\infty }}{a}_n{b}_n$.

Consider Divergence Argument

Suppose $b_n=1$ for all $n$, then $\sum _{n=1}^{\mathrm{\infty }}{a}_n{b}_n$ is exactly $\sum _{n=1}^{\mathrm{\infty }}{a}_n$, which converges by the problem's condition. Now consider when $b_n$ takes any pattern of 0s and 1s that ensures an infinite number of terms remain in $\sum _{n=1}^{\mathrm{\infty }}{a}_n{b}_n$.

Apply Limit Comparison Test

Check if there exists a subsequence of all non-zero terms in $b_n$ such that its corresponding subsequence of $a_n$ in $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges. While each individual component $a_n{b}_n$ might still go to zero, this does not itself ensure convergence of the series without further condition.

Conclusion of Convergence

The convergence of the original series implies each $b_n=0$ or function asymptotically captures 0 through few non-zero terms in the modification. Since every AED vanishes in the pounds $b_n$ scenario, the series converges when $b_n$ isn't erratically non-zero in each $n$, and $\sum a_n$ as $\sum a_n{b}_n$ captures consistently-asymptotic across $n\in b_n$.

Key concepts

Convergent Series

A convergent series is a fundamental concept in calculus and analysis. It involves an infinite sum of terms that approaches a specific value. When we say a series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges, it means that as we keep adding more and more terms, the total sum gets closer and closer to a fixed number.
The key condition for a series to converge is that the terms $a_n$ must get smaller and approach zero as $n$ becomes very large. In mathematical terms, we say $a_n\to 0$ as $n\to \mathrm{\infty }$.
- For example, the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$ converges to a specific value, because its terms become very tiny as $n$ grows.
- On the other hand, the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$ does not converge, because the terms do not decrease fast enough.
Understanding when a series converges is crucial for various applications in mathematics and science, as it ensures predictability and stability in computations.

Sequence of Zeros and Ones

A sequence of zeros and ones is a simple yet powerful concept in mathematics. In our problem, each element $b_n$ of the sequence is either 0 or 1.
This type of sequence can dictate which terms in another series matter by multiplying terms by 0 or 1. When a term is multiplied by 0, it effectively "turns off" that term, making it disappear from the sum. Conversely, multiplying by 1 means the term is included.
- Such sequences can create various scenarios, ranging from sequences where all terms are zero (making the entire sum zero), or sequences with just some ones scattered, affecting which terms from $a_n$ contribute to the sum.
- These sequences are essential in problems involving conditional sums, switch functions, and in binary systems used in computing.
The randomness or pattern of ones and zeros can significantly influence whether the modified series maintains convergence, diverges, or flips between states depending on the sequence's structure.

Limit Comparison Test

The Limit Comparison Test is a handy tool for determining the convergence or divergence of series. It provides a way to compare two series' behaviors by examining the limits of their term ratios.
To apply this test to series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ and $\sum _{n=1}^{\mathrm{\infty }}{b}_n$:
- Calculate the limit $\underset{n\to \mathrm{\infty }}{lim}\frac{a_n}{b_n}$.
If the limit is a positive finite number, both series converge or diverge together. This test is particularly useful when $a_n$ is similar to $b_n$ for large $n$, allowing us to draw conclusions about the convergence of more complex or modified series.
In our context, by checking subsequences made from ones in $b_n$, you can use the Limit Comparison Test to determine if these subsets $a_n{b}_n$ still behave in a way that guarantees convergence. However, since $b_n$ could be unpredictable, this process requires careful consideration of every subsequence that remains potentially significant in sum aggregation.