To prove that the sequence

$a_n = \sum_{k=1}^{n} \frac{n}{n^2 + k}$

is convergent using the sandwich (squeeze) theorem, we need to find two sequences, say $b_n$ and $c_n$, such that:

$b_n \leq a_n \leq c_n \quad \text{for all } n$

and both $b_n$ and $c_n$ converge to the same limit as $n \to \infty$. Here's how we proceed:

Step 1: Express the General Term

The given sequence $a_n$ is a sum of terms of the form

$\frac{n}{n^2 + k}, \quad k = 1, 2, \dots, n$

Let's analyze the behavior of each individual term. For large $n$, we approximate:

$\frac{n}{n^2 + k} \approx \frac{n}{n^2} = \frac{1}{n}$

Thus, for large $n$, each term behaves like $\frac{1}{n}$.

Step 2: Find Upper Bound $c_n$

Since $k \geq 1$, we have:

$n^2 + k \geq n^2 + 1$

Thus,

$\frac{n}{n^2 + k} \leq \frac{n}{n^2 + 1}$

Summing this inequality over all $k$ from 1 to $n$, we get:

$a_n = \sum_{k=1}^{n} \frac{n}{n^2 + k} \leq \sum_{k=1}^{n} \frac{n}{n^2 + 1} = n \cdot \frac{n}{n^2 + 1}$

Simplifying the right-hand side:

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

For large $n$, $\frac{n^2}{n^2 + 1} \to 1$.

Thus, we have an upper bound:

$a_n \leq 1$

Step 3: Find Lower Bound $b_n$

Similarly, since $k \leq n$, we have:

$n^2 + k \leq n^2 + n$

Thus,

$\frac{n}{n^2 + k} \geq \frac{n}{n^2 + n} = \frac{n}{n(n + 1)} = \frac{1}{n + 1}$

Summing this inequality over all $k$ from 1 to $n$, we get:

$a_n = \sum_{k=1}^{n} \frac{n}{n^2 + k} \geq \sum_{k=1}^{n} \frac{1}{n + 1} = n \cdot \frac{1}{n + 1}$

Simplifying the right-hand side:

$a_n \geq \frac{n}{n + 1}$

For large $n$, $\frac{n}{n + 1} \to 1$.

Thus, we have a lower bound:

$a_n \geq 1 - \frac{1}{n + 1}$

Step 4: Apply the Sandwich (Squeeze) Theorem

Now, we have the inequalities:

$1 - \frac{1}{n + 1} \leq a_n \leq \frac{n^2}{n^2 + 1}$

As $n \to \infty$:

$1 - \frac{1}{n + 1} \to 1 \quad \text{and} \quad \frac{n^2}{n^2 + 1} \to 1$

By the sandwich theorem, we conclude that:

$a_n \to 1 \quad \text{as} \quad n \to \infty$

Thus, the sequence $a_n$ is convergent, and its limit is $1$.

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Would you like more details or have any questions? Here are 5 related questions to expand your understanding:

1. How does the behavior of the general term $\frac{n}{n^2 + k}$ affect the convergence?
2. Can the sandwich theorem be applied to sequences involving sums in more complex cases?
3. How do we find the upper and lower bounds in different convergence problems?
4. Why is it important that both bounds converge to the same limit in the sandwich theorem?
5. How would this proof change if the terms were $\frac{n}{n^2 + k^2}$?

Tip: When using the sandwich theorem, the key is finding bounds that closely approximate the behavior of the sequence for large values of $n$.