Prove that the sequence n/(n+1) is convergent.
Understand the Problem
This question is asking to prove the convergence of the sequence defined by the term n/(n+1). Convergence in this context refers to whether the sequence approaches a specific limit as n approaches infinity.
Answer
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Answer for screen readers
The sequence ( a_n = \frac{n}{n+1} ) converges to 1 as ( n ) approaches infinity.
Steps to Solve
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Identify the Sequence The sequence we are examining is given by the formula ( a_n = \frac{n}{n+1} ).
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Simplify the Expression To analyze the behavior of this sequence as ( n ) approaches infinity, simplify ( a_n ): $$ a_n = \frac{n}{n+1} = \frac{n}{n(1 + \frac{1}{n})} = \frac{1}{1 + \frac{1}{n}} $$
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Take the Limit as n Approaches Infinity Now, find the limit of ( a_n ) as ( n ) approaches infinity: $$ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{1 + \frac{1}{n}} $$ As ( n ) tends to infinity, ( \frac{1}{n} ) approaches 0.
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Evaluate the Limit Therefore, the limit becomes: $$ \lim_{n \to \infty} a_n = \frac{1}{1 + 0} = 1 $$
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Conclusion about Convergence Since the limit exists and equals 1, we conclude that the sequence ( a_n = \frac{n}{n+1} ) converges to 1.
The sequence ( a_n = \frac{n}{n+1} ) converges to 1 as ( n ) approaches infinity.
More Information
This result shows that as ( n ) becomes very large, the fraction ( \frac{n}{n+1} ) gets closer and closer to 1. This is a classic example of a convergent sequence in mathematics.
Tips
- Confusing convergence with divergence: Ensure that you are looking for the limit as ( n ) approaches infinity, not the initial terms of the sequence.
- Forgetting to simplify the terms: Always check if the sequence can be simplified to make limit calculation easier.
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