Problem 10 Prove that if \(\left\\{a_{n}\ri... [FREE SOLUTION]

Chapter 13: Problem 10

Prove that if \(\left\\{a_{n}\right\\}\) converges to \(L\) and \(\left\\{b_{n}\right\\}\) converges to \(M,\) then the sequence \(\left\\{a_{n}+b_{n}\right\\}\) converges to \(L+M\)

Short Answer

Expert verified

The sequence \(\left\{a_{n}+b_{n}\right\}\) converges to \(L+M\).

Step by step solution

Understand the Definition of Convergence

First, it is important to understand what it means for a sequence to converge. A sequence \(\left\{a_{n}\right\}\) is said to converge to \(L\) if, for every positive real number \(\epsilon > 0\), there exists a natural number \(N\) such that for all \(n>N\), the absolute value \(|a_{n}-L| < \epsilon\). Similarly for the sequence \(\left\{b_{n}\right\}\) converging to \(M\).

Apply Definition to Both Sequences

Given that \(\left\{a_{n}\right\}\) converges to \(L\) and \(\left\{b_{n}\right\}\) converges to \(M\), for every \(\epsilon > 0\), there exist natural numbers \(P\) and \(Q\) such that for all \(n>P\) we have \(|a_{n}-L| < \frac{\epsilon}{2}\) and for all \(n>Q\) we have \(|b_{n}-M| < \frac{\epsilon}{2}\). Choose \(N = \max\{P, Q\}\).

Combine and Show New Sequence Convergence

We consider the sequence \( \left\{a_{n}+b_{n}\right\}\). For all \(n>N\), we have \(|(a_{n}+b_{n})-(L+M)| = |(a_{n}-L)+(b_{n}-M)| \leq |a_{n}-L|+|b_{n}-M| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon\). Thus, the sequence \(\left\{a_{n}+b_{n}\right\}\) converges to \(L+M\).

Summarize The Conclusion

In conclusion, if a sequence \(\left\{a_{n}\right\}\) converges to \(L\) and another sequence \(\left\{b_{n}\right\}\) converges to \(M,\) then the sequence formed by the term-wise sum, \(\left\{a_{n}+b_{n}\right\}\), converges to the sum of the limits, \(L+M\). This is often referred to as the sum rule for convergent sequences.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Convergence

Sequence convergence is a fundamental concept in calculus and analysis. When we say a sequence converges, we mean that as you go further and further along the sequence, the numbers in the sequence get closer and closer to a specific value, called the limit. This process is like traveling along a path that ends at a target point. For the sequence \(\{a_{n}\}\) to converge to a limit \(L\), we need:

For any small positive number \(\epsilon\), there exists a point in the sequence after which all terms are within \(\epsilon\) of \(L\).
If you choose a neighborhood around \(L\) (within \(\epsilon\)), the terms of the sequence eventually stay inside this neighborhood.

This concept helps to ensure that sequences have predictable long-term behavior, even if the initial terms vary significantly from the limit.

Sum Rule

The sum rule for sequences is a handy tool for combining convergent sequences. It states that if you have two sequences \(\{a_{n}\}\) and \(\{b_{n}\}\), where each one individually converges to a limit \(L\) and \(M\) respectively, then the sequence formed by adding their corresponding terms, \(\{a_{n} + b_{n}\}\), will converge to \(L + M\).

This rule follows from the arithmetic properties of limits, where you can distribute limits over sums. Here's why:

Both sequences approach their respective limits as \(n\) grows larger.
Adding the small differences \(|a_{n} - L|\) and \(|b_{n} - M|\) results in a value that's also small, which ensures \(|a_{n} + b_{n} - (L + M)|\) remains small.

This property is useful for simplifying complex limit problems in calculus and provides a way to combine solutions.

Definition of Convergence

Understanding the definition of convergence is crucial to grasping how sequences behave as they progress. To say that a sequence \(\{a_{n}\}\) converges to a limit \(L\), mathematically means:

For every \(\epsilon > 0\), there must be some natural number \(N\) such that anytime the sequence vertex \(n\) is greater than \(N\), the terms \(a_{n}\) remain within this tolerance \(\epsilon\) around \(L\).
Said simply, after a certain point, all terms of the sequence are locked in a close range from \(L\).

This definition emphasizes that convergence is all about eventual consistency, showing that the sequence moves closer and stays close to a specific target value, even if the starting numbers deviate. This framework is essential for ensuring clarity and rigor in mathematical arguments involving limits and convergence.

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91影视

Prove that if \(\left\\{a_{n}\right\\}\) converges to \(L\) and \(\left\\{b_{n}\right\\}\) converges to \(M,\) then the sequence \(\left\\{a_{n}+b_{n}\right\\}\) converges to \(L+M\)

Short Answer

Step by step solution

Understand the Definition of Convergence

Apply Definition to Both Sequences

Combine and Show New Sequence Convergence

Summarize The Conclusion

Key Concepts

Sequence Convergence

Sum Rule

Definition of Convergence

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