# Does the Sequence a_n Converge or Diverge as n Approaches Infinity?

In summary, the conversation discusses determining whether the sequence a_n = \frac{1^2}{n^3} + \frac{2^2}{n^3} + ... + \frac{n^2}{n^3} converges or diverges and how to find its limit if it does converge. The conversation explores various methods such as comparing it to 1/n and using the p-series test, but ultimately concludes that the number of terms in the sum also increases to infinity, making it difficult to determine the exact value of the sum. Ultimately, the conversation encourages finding the solution independently rather than simply providing an answer.
Determine whether the sequence $$a_n = \frac{1^2}{n^3} + \frac{2^2}{n^3} + ... + \frac{n^2}{n^3}$$ converges or diverges. If it converges, find the limit.

wouldnt it converge to 0?

Maybe, maybe not. Why do you think it will go to 0? Any ideas on how to go about trying to prove it?

That seems like a difficult problem, since you have a sequence of series.

The first few terms in the sequence would be:

$$\left{1, \frac{1^2}{2^3} + \frac{2^2}{2^3}, \frac{1^2}{3^3} + \frac{2^2}{3^3} + \frac{3^2}{3^3}, \frac{1^2}{4^3} + \frac{2^2}{4^3} + \frac{3^2}{4^3} + \frac{4^2}{4^3},...\right}$$

The nth term is increasing and bounded by one

$$a_n \leq \frac{n^2}{n^3} + \frac{n^2}{n^3} + ... + \frac{n^2}{n^3} = n\frac{n^2}{n^3} = 1$$

Actually, the nth term can be rewritten as

$$a_n = \frac{1}{n^3}\sum_{i=1}^{n}i^2$$

Do you recognize this sum?

The reason why the value of convergence of this sequence is not simply obtained by doing

$$\lim_{n\rightarrow \infty} \frac{1^2}{n^3} + \frac{2^2}{n^3} + ... + \frac{n^2}{n^3} = \lim_{n\rightarrow \infty} \frac{1^2}{n^3} + \lim_{n\rightarrow \infty} \frac{2^2}{n^3} + ... + \lim_{n\rightarrow \infty} \frac{n^2}{n^3} = 0+0+...+0$$

is that the NUMBER OF TERMS in the sum augment to infinity as well! So you can't be sure what the sum's going to be: even though all members of the sum go to zero, there is an infinity of them.

so it converges, but no answer?

just a thought, wouldn't the limit comparison test work on this if you choose your b_n to be 1/n, and use the p-series to say b_n diverges?

Pro: we're here to help, not give answers. You should have enough information to figure it out yourself.

mug: what are you comparing to 1/n?

## 1. What does it mean for a series to converge or diverge?

Convergence and divergence refer to the behavior of a series, which is a sequence of numbers being added together. A series is said to converge if the sum of its terms approaches a finite number as the number of terms increases. Conversely, a series is said to diverge if the sum of its terms increases without bound as the number of terms increases.

## 2. How can you determine if a series converges or diverges?

There are several tests that can be used to determine if a series converges or diverges, including the comparison test, the ratio test, and the integral test. These tests involve comparing the given series to a known series or using mathematical calculations to determine the behavior of the series.

## 3. What is the significance of knowing if a series converges or diverges?

Knowing whether a series converges or diverges is important for understanding the behavior of the sequence of numbers. It can also have practical applications in various fields, such as physics, engineering, and economics, where series are used to model real-world phenomena.

## 4. Can a series both converge and diverge?

No, a series can only have one of two behaviors: convergence or divergence. However, it is possible for different series to have different convergence or divergence behaviors, even if they have the same terms.

## 5. How does the rate of convergence or divergence affect the series?

The rate of convergence or divergence refers to how quickly the sum of the terms approaches a finite number or how quickly the sum of the terms increases without bound. A series with a faster rate of convergence will approach its limit more quickly, while a series with a slower rate of convergence will take longer to approach its limit. Similarly, a series with a faster rate of divergence will increase more quickly, while a series with a slower rate of divergence will increase more slowly.

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