https://r2.erweima.ai/mathbot/image/1726993521858u2y8ax7l.jpg

The mathematical expression from the image asks to prove the following limit:

\[
\lim_{n \to \infty} \left[ \frac{1}{(n+1)^3} + \frac{2}{(n+2)^3} + \cdots + \frac{n}{(2n)^3} \right] = 0
\]

### Solution Outline:

1. **Sum Expression**: The sum inside the limit represents a series with terms of the form \( \frac{k}{(n+k)^3} \), where \( k = 1, 2, \dots, n \).
   
2. **Estimate Each Term**: For large \( n \), the general term \( \frac{k}{(n+k)^3} \) can be approximated as:
   \[
   \frac{k}{(n+k)^3} \approx \frac{k}{n^3(1 + \frac{k}{n})^3}.
   \]
   For large \( n \), the denominator grows rapidly, making each term very small.

3. **Upper Bound of the Sum**: As \( n \to \infty \), each term decays at a rate proportional to \( \frac{1}{n^2} \). Since there are \( n \) terms, the overall sum behaves like \( \frac{1}{n} \), which approaches 0 as \( n \to \infty \).

4. **Conclusion**: Since the sum of all the terms tends to zero, the limit is indeed 0.

### Detailed Steps:

1. Express the sum in the form:
   \[
   \sum_{k=1}^{n} \frac{k}{(n+k)^3}.
   \]

2. Approximate the denominator for large \( n \) and simplify the sum.

3. Use bounding techniques or the integral approximation method to show that the sum converges to 0 as \( n \to \infty \).

Would you like a detailed explanation of the bounding and estimation process, or help with a specific part of this problem?

### Relative Questions:
1. How do we rigorously estimate the decay rate of the terms in the sum?
2. Can we use the integral approximation to estimate sums like this one?
3. How do we deal with sums that involve terms like \( \frac{k}{(n+k)^3} \)?
4. What role does asymptotic analysis play in this limit?
5. What other examples of sums involving cubes could be solved similarly?

#### Tip:
When handling limits of sums, breaking down the behavior of individual terms and using approximation techniques can greatly simplify the problem.

Prove that the limit: lim (n -> infinity) [1/(n+1)^3 + 2/(n+2)^3 + ... + n/(2n)^3] = 0.

This proof shows how to calculate the limit as n approaches infinity of a series involving cubic denominators. The sum involves terms of the form k/(n+k)^3, and the solution includes asymptotic analysis and bounding techniques. Ideal for university-level students studying calculus and series.

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