Given Two Convergent Sequences Find Lim ⁡ N → ∞ A N N + B N N + 2018 N \lim_{n\to\infty}\sqrt[n]{a_n^n+b_n^n+2018} Lim N → ∞ ​ N A N N ​ + B N N ​ + 2018 ​

Introduction

In this article, we will explore the concept of convergent sequences and how to find the limit of a root expression involving these sequences. Given two convergent sequences, $a_n$ and $b_n$, we are tasked with finding the limit of the expression $\lim_{n\to\infty}\sqrt[n]{a_n^n+b_n^n+2018}$.

Understanding Convergent Sequences

A convergent sequence is a sequence that approaches a finite limit as the index of the sequence approaches infinity. In other words, a sequence $a_n$ is said to converge to a limit $a$ if for every positive real number $\epsilon$, there exists a positive integer $N$ such that for all $n > N$, $|a_n - a| < \epsilon$.

In this case, we are given two convergent sequences, $a_n$ and $b_n$, that converge to limits $a$ and $b$, respectively, where both $a$ and $b$ are greater than 1.

The Limit of a Root Expression

The given expression involves the root of a sum of two terms, each raised to the power of $n$. To find the limit of this expression, we can start by analyzing the behavior of the terms inside the root.

As $n$ approaches infinity, both $a_n^n$ and $b_n^n$ approach infinity as well, since $a_n$ and $b_n$ converge to positive limits. Therefore, the sum $a_n^n + b_n^n$ also approaches infinity.

Using the Binomial Theorem

To analyze the behavior of the root expression, we can use the binomial theorem to expand the expression $(a_n^n + b_n^n + 2018)^{1/n}$.

Using the binomial theorem, we can write:

$$(a_n^n + b_n^n + 2018)^{1/n} = (a_n^n)^{1/n} + (b_n^n)^{1/n} + 2018^{1/n} + \ldots$$

As $n$ approaches infinity, the terms $(a_n^n)^{1/n}$ and $(b_n^n)^{1/n}$ approach $a$ and $b$, respectively, since $a_n$ and $b_n$ converge to $a$ and $b$.

The Dominant Term

The term $2018^{1/n}$ approaches 1 as $n$ approaches infinity, since $2018$ is a constant. Therefore, this term becomes negligible compared to the other terms.

The Limit of the Root Expression

Using the results from the previous sections, we can now find the limit of the root expression.

As $n$ approaches infinity, the terms $(a_n^n)^{1/n}$ and $(b_n^n)^{1/n}$ approach $a$ and $b$, respectively. Therefore, the limit of the root expression is:

$$\lim_{n\to\infty}\sqrt[n]{a_n^n+b_n^n+2018} = a + b$$

Conclusion

In this article, we have explored the concept of convergent sequences and how to find the limit of a root expression involving these sequences. We have shown that the limit of the root expression is equal to the sum of the limits of the two convergent sequences.

Example

To illustrate this result, let's consider an example. Suppose we have two convergent sequences, $a_n = 1 + 1/n$ and $b_n = 1 + 1/n$, that converge to limits $a = 1$ and $b = 1$, respectively.

Using the result from this article, we can find the limit of the root expression:

$$\lim_{n\to\infty}\sqrt[n]{a_n^n+b_n^n+2018} = 1 + 1 = 2$$

References

• [1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
• [2] Bartle, R. G. (1976). The Elements of Real Analysis. Wiley.

Further Reading

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about convergent sequences and the limit of a root expression.

Q: What is a convergent sequence?

A: A convergent sequence is a sequence that approaches a finite limit as the index of the sequence approaches infinity. In other words, a sequence $a_n$ is said to converge to a limit $a$ if for every positive real number $\epsilon$, there exists a positive integer $N$ such that for all $n > N$, $|a_n - a| < \epsilon$.

Q: How do I know if a sequence is convergent?

A: To determine if a sequence is convergent, you can use the following criteria:

• The sequence is bounded, meaning that there exists a real number $M$ such that $|a_n| \leq M$ for all $n$.
• The sequence is monotonic, meaning that it is either increasing or decreasing.
• The sequence has a limit, meaning that it approaches a finite value as $n$ approaches infinity.

Q: What is the limit of a root expression?

A: The limit of a root expression is the value that the expression approaches as the index of the sequence approaches infinity. In the case of the given expression, $\lim_{n\to\infty}\sqrt[n]{a_n^n+b_n^n+2018}$, the limit is equal to the sum of the limits of the two convergent sequences, $a$ and $b$.

Q: How do I find the limit of a root expression?

A: To find the limit of a root expression, you can use the following steps:

1. Determine if the sequences $a_n$ and $b_n$ are convergent.
2. Find the limits of the sequences $a_n$ and $b_n$.
3. Use the result from this article to find the limit of the root expression.

Q: What if the sequences $a_n$ and $b_n$ are not convergent?

A: If the sequences $a_n$ and $b_n$ are not convergent, then the limit of the root expression may not exist. In this case, you may need to use other methods to analyze the behavior of the expression.

Q: Can I use this result to find the limit of other root expressions?

A: Yes, this result can be used to find the limit of other root expressions. Simply apply the same steps as before to determine if the sequences are convergent, find the limits of the sequences, and use the result from this article to find the limit of the root expression.

Q: Are there any other applications of this result?

A: Yes, this result has many applications in mathematics and other fields. For example, it can be used to analyze the behavior of functions, solve equations, and model real-world phenomena.

Conclusion

In this article, we have answered some of the most frequently asked questions about convergent sequences and the limit of a root expression. We hope that this information has been helpful in understanding this important result in mathematics.