Evaluate the following limits :
`Lim_(n to oo) (1+2+3+...+n)/(n^(2)) ( or Lim_(x to oo) (Sigman)/(n^(2)))`

Answer

To evaluate the limit

\[ \lim_{n \to \infty} \frac{1 + 2 + 3 + \ldots + n}{n^2} \]

we can follow these steps:

### Step 1: Use the formula for the sum of the first \( n \) natural numbers
The sum of the first \( n \) natural numbers is given by the formula:

\[ S_n = \frac{n(n + 1)}{2} \]

### Step 2: Substitute the sum into the limit
Now, we can substitute this formula into our limit:

\[ \lim_{n \to \infty} \frac{S_n}{n^2} = \lim_{n \to \infty} \frac{\frac{n(n + 1)}{2}}{n^2} \]

### Step 3: Simplify the expression
This simplifies to:

\[ \lim_{n \to \infty} \frac{n(n + 1)}{2n^2} = \lim_{n \to \infty} \frac{n^2 + n}{2n^2} \]

### Step 4: Factor out \( n^2 \) from the numerator
We can factor out \( n^2 \) from the numerator:

\[ \lim_{n \to \infty} \frac{n^2(1 + \frac{1}{n})}{2n^2} \]

### Step 5: Cancel \( n^2 \) in the numerator and denominator
Now, we can cancel \( n^2 \):

\[ \lim_{n \to \infty} \frac{1 + \frac{1}{n}}{2} \]

### Step 6: Evaluate the limit
As \( n \) approaches infinity, \( \frac{1}{n} \) approaches 0:

\[ \lim_{n \to \infty} \frac{1 + 0}{2} = \frac{1}{2} \]

### Final Answer
Thus, the limit is:

\[ \frac{1}{2} \]

---