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Chapter 8: Problem 13

Determine whether the sequence converges or diverges. $$a_{n}=\frac{n^{2}+1}{n+1}$$

Short Answer

Expert verified
The sequence diverges.

Step by step solution

01

Simplify the Sequence

To simplify the sequence, divide the numerator and denominator of the fraction by \(n^{2}\). This leads to \[a_{n}=\frac{n^{2}+1}{n+1}=\frac{1+\frac{1}{n^{2}}}{\frac{1}{n}+\frac{1}{n^{2}}}\]
02

Take the Limit

Now take the limit of the sequence as \(n\) approaches infinity: \[\lim_{n \to \infty}\frac{1+\frac{1}{n^{2}}}{\frac{1}{n}+\frac{1}{n^{2}}}\] The terms \(\frac{1}{n^{2}}\) and \(\frac{1}{n}\) will approach zero as \(n\) approaches infinity.
03

Find the Limit and Check Convergence

Substitute the limits into the sequence: \[\lim_{n \to \infty}\frac{1+0}{0+0}= \infty\] The limit is not finite, therefore, the sequence diverges.

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Key Concepts

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

Limit of a Sequence
When we talk about the limit of a sequence, we are interested in finding out what value, if any, the elements of the sequence approach as the sequence progresses towards infinity. In mathematical terms, for a sequence \(a_n\), the limit is a value \(L\) such that for every small number \(\epsilon > 0\), there exists a number \(N\) where for all \(n > N\), the difference \(|a_n - L| < \epsilon\). This essentially means that as \(n\) becomes very large, \(a_n\) gets very close to \(L\).

Determining the limit helps us know if the sequence settles down to a particular value or not. In the case of our exercise, after simplifying the sequence by dividing by \(n^2\), we attempted to find the limit as \(n\) became very large. The idea is to test if the sequence stabilizes, thereby converging to a limit.
Divergence of Sequences
A sequence diverges when it does not converge to a specific value as it progresses towards infinity. Divergence can take many forms, such as growing larger and larger without bound, oscillating between values, or becoming undefined.

In our exercise, after simplifying the sequence \(a_n\), we found that the limit as \(n\) approaches infinity resulted in an undefined form, specifically \(\infty\). Since we did not find a finite number to which the sequence stabilizes or converges, it shows the divergence of the sequence. A diverging sequence implies that no matter how far we extend, the sequence will not settle into a single value.
Infinite Limits
Infinite limits occur when the value of a sequence increases or decreases without bound as it progresses. This usually results in the value approaching \(\infty\) or \(-\infty\).

In our specific problem, after breaking down the sequence into simpler terms and then evaluating the limit, we saw that it tended towards \(\infty\). This means that as we go further out in the sequence, the terms increase indefinitely. The term infinite limit indicates that as \(n\) becomes larger, the sequence's values continue to grow, demonstrating the absence of convergence.

Infinite limits are crucial in understanding that not all sequences settle into a particular number and can help in identifying the behavior of a divergent sequence.

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