To evaluate the limit

$\lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{n^2},$

we start by simplifying the numerator. The sum of the first $n$ natural numbers is given by the formula:

$1 + 2 + 3 + \dots + n = \frac{n(n + 1)}{2}.$

Substituting this into the expression:

$\lim_{n \to \infty} \frac{\frac{n(n + 1)}{2}}{n^2}.$

Simplify the fraction:

$\lim_{n \to \infty} \frac{n(n + 1)}{2n^2}.$

Distribute $n$ in the numerator:

$\lim_{n \to \infty} \frac{n^2 + n}{2n^2}.$

Separate the terms in the fraction:

$\lim_{n \to \infty} \frac{n^2}{2n^2} + \frac{n}{2n^2}.$

Simplify each term:

$\lim_{n \to \infty} \frac{1}{2} + \frac{1}{2n}.$

As $n \to \infty$, the term $\frac{1}{2n} \to 0$, so the limit becomes:

$\frac{1}{2} + 0 = \frac{1}{2}.$

Final Answer:

$\lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{n^2} = \frac{1}{2}.$

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Related Questions:

1. What is the significance of the result $\frac{1}{2}$ in this context?
2. How does the formula for the sum of the first $n$ natural numbers relate to other series?
3. How would the result change if the denominator were $n^3$ instead of $n^2$?
4. Can this limit be interpreted geometrically?
5. How does this approach generalize to other power series limits?

Tip:

When working with limits involving sums, look for well-known formulas to simplify the problem efficiently!