Problem 91 - Essential Calculus

Question

In Exercises 91-94, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\left\\{a_{n}\right\\}\) converges to 3 and \(\left\\{b_{n}\right\\}\) converges to 2 , then \(\left\\{a_{n}+b_{n}\right\\}\) converges to 5 .

Step-by-Step Solution

Answer

The statement is true. If \(\left\{a_{n}\right\}\) converges to 3 and \(\left\{b_{n}\right\}\) converges to 2, then indeed \(\left\{a_{n}+b_{n}\right\}\) converges to 5, in accordance with the rules of sequence convergence.

Step 1: Understanding Sequence Convergence

In mathematics, a series or sequence converges if its terms approach a specific value as they progress to infinity. In this case, the sequences \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\) are already established to converge to 3 and 2 respectively. Convergence ensures that the values in a sequence become closer to a certain value, known as the limit, as the number of terms increase.

Step 2: Understanding the Sum of Convergent Sequences

It is known from algebra that if two sequences converge, then the sequence of their sums also converges. Specifically, if \(\left\{a_{n}\right\}\) converges to A and \(\left\{b_{n}\right\}\) converges to B, then the sequence \(\left\{a_{n}+b_{n}\right\}\) will converge to \(A+B\).

Step 3: Applying the Convergence Rule to the Given Sequences

In the current scenario, \(\left\{a_{n}\right\}\) converges to 3 and \(\left\{b_{n}\right\}\) converges to 2. By applying the convergence rule outlined above, the sequence \(\left\{a_{n}+b_{n}\right\}\) should therefore converge to \(3+2 = 5\).

Key Concepts

Understanding Sequence Convergence

Sequence convergence is a fundamental concept in mathematics, especially in the fields of calculus and analysis. A sequence is simply an ordered list of numbers, and the convergence of a sequence refers to the behavior of the sequence as its terms progress toward infinity. If a sequence converges, this means that as you move through the sequence (from the first term and beyond), the terms get closer and closer to a single number, known as the limit of the sequence.

For instance, if we consider a sequence \(\{a_n\}\), and we say that it converges to the number L, we imply that the terms of the sequence, \(a_1, a_2, a_3, ...\), become arbitrarily close to L as n becomes large. Mathematically speaking, for every positive number \(\epsilon\) no matter how small, there exists a positive integer N such that for all \(n \geq N\), the distance between \(a_n\) and L is less than \(\epsilon\). This property ensures a predictable long-term behavior, despite the initial terms of the sequence potentially exhibiting erratic behavior.

It's important for students to grasp that this idea of getting 'arbitrarily close' does not necessarily mean the sequence ever reaches the limit. It means that after a certain point, the terms of the sequence are indistinguishable from the limit for all practical purposes.

Sum of Convergent Sequences

When we have two sequences that converge, a natural question to ask is what happens when we combine them. For example, if we add them together, term by term, does the resulting sequence also converge? The answer is yes. The sum of two convergent sequences is itself a convergent sequence.

To understand why this is true, let's think about what it means for each sequence to converge. Say we have sequences \(\{a_n\}\) and \(\{b_n\}\) which converge to A and B respectively. This means that for any small number \(\epsilon > 0\), there is a point in each sequence beyond which all terms are within \(\epsilon\) of their respective limits A and B.

If we add the terms of these sequences together to form a new sequence \(\{a_n + b_n\}\), the terms of this new sequence will eventually be within \(2\epsilon\) (or any other arbitrary small margin you choose) of the number \(A + B\). This is due to the properties of limits and arithmetic: limits can be added together just like regular numbers, and the closeness to these limits is preserved under addition. Hence, the convergence of the sum follows logically from the convergence of the individual sequences.

Limit of a Sequence

The limit of a sequence is perhaps the most crucial aspect when discussing convergence. The limit represents the value that the terms of a sequence are approaching as the sequence progresses towards infinity. It is the value 'L' in our previous discussion on sequence convergence.

A limit is a way of formalizing the idea that something gets closer and closer to a particular value without necessarily ever being that value. Limits allow mathematicians to discuss and handle infinity in a precise way, rather than just saying a sequence 'goes on forever.' We can talk about the behavior of a sequence as it goes on forever by discussing its limit.

For a sequence \(\{a_n\}\) converging to a limit L, the notation \(\lim_{n \to \infty} a_n = L\) is used. This is read as 'the limit of the sequence \(a_n\) as n approaches infinity is L.' This expression captures the essence of convergence in a succinct form. When dealing with sequences, always remember that a limit is a theoretical concept. We can't actually reach infinity to check the value of a sequence at infinity, but we can predict and calculate where it’s headed using the concept of a limit.