\lim_{n\rightarrow\infty}\left[\frac{1}{n^{2}}\sum_{k=1}^{n} k\ln(n+k)-\frac{n+1}{2 n}\ln n\right]

We are tasked with evaluating the limit:

\[
\lim_{n\rightarrow\infty}\left[\frac{1}{n^{2}}\sum_{k=1}^{n} k\ln(n+k)-\frac{n+1}{2 n}\ln n\right]
\]

Let's break down this expression step by step:

### 1. Analyze the Sum \(\frac{1}{n^2} \sum_{k=1}^{n} k \ln(n+k)\)

We first examine the sum:
\[
S_n = \frac{1}{n^2} \sum_{k=1}^{n} k \ln(n+k)
\]

For large \(n\), we approximate \(\ln(n+k)\). Since \(k\) is much smaller than \(n\) when \(n\) is large, we have:
\[
\ln(n+k) \approx \ln n + \ln\left(1 + \frac{k}{n}\right)
\]

Using the approximation \(\ln(1+x) \approx x\) for small \(x\), we get:
\[
\ln(n+k) \approx \ln n + \frac{k}{n}
\]

Substituting this back into the sum:
\[
S_n \approx \frac{1}{n^2} \sum_{k=1}^{n} k \left(\ln n + \frac{k}{n}\right)
\]
This simplifies to:
\[
S_n \approx \frac{1}{n^2} \left[\ln n \sum_{k=1}^{n} k + \frac{1}{n} \sum_{k=1}^{n} k^2\right]
\]
We know the following sums:
\[
\sum_{k=1}^{n} k = \frac{n(n+1)}{2}, \quad \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}
\]

Substituting these into the expression for \(S_n\):
\[
S_n \approx \frac{1}{n^2} \left[\ln n \cdot \frac{n(n+1)}{2} + \frac{1}{n} \cdot \frac{n(n+1)(2n+1)}{6}\right]
\]

For large \(n\), this becomes:
\[
S_n \approx \frac{\ln n}{2} + \frac{1}{3}
\]

### 2. Analyze the Term \(\frac{n+1}{2n} \ln n\)

This term can be simplified as \(n \to \infty\):
\[
\frac{n+1}{2n} \ln n \approx \frac{1}{2} \ln n
\]

### 3. Putting It All Together

Now, subtract the two expressions:
\[
S_n - \frac{n+1}{2n} \ln n \approx \left(\frac{\ln n}{2} + \frac{1}{3}\right) - \frac{1}{2} \ln n
\]
Simplifying:
\[
S_n - \frac{n+1}{2n} \ln n \approx \frac{1}{3}
\]

### Conclusion

Thus, the limit is:
\[
\lim_{n \to \infty} \left[\frac{1}{n^{2}}\sum_{k=1}^{n} k\ln(n+k)-\frac{n+1}{2 n}\ln n\right] = \frac{1}{3}
\]

Would you like further details on any part of this process?

Here are 5 related questions you might consider:
1. How would this change if the logarithmic term was \(\ln(n^2+k)\)?
2. Can you generalize this approach for sums involving higher powers of \(k\)?
3. How does this problem connect to the asymptotics of sums and series?
4. What is the effect of using other approximations for \(\ln(1 + x)\)?
5. How could we approach the same problem using numerical methods?

**Tip:** In dealing with sums and limits, understanding approximation techniques like Taylor expansions can significantly simplify complex expressions.

Evaluate the limit of a summation involving logarithms and series as n approaches infinity. This advanced calculus problem uses asymptotic analysis, Taylor expansion, and summation formulas. Detailed step-by-step solution for university-level students.

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