The value of `lim_(n->oo)(sqrt(1)+sqrt(2)+sqrt(3)+.....+sqrt(n))/(nsqrt(n))` is

Options

A. `2//3`
B. `1//2`
C. `1//3`
D. `3//2`

Answer

To solve the limit

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

we can approach it using the concept of Riemann sums, which can be interpreted as integrals.

### Step 1: Rewrite the Sum
The numerator can be expressed as a summation:

\[ \sum_{r=1}^{n} \sqrt{r}. \]

Thus, we can rewrite our limit as:

\[ \lim_{n \to \infty} \frac{1}{n \sqrt{n}} \sum_{r=1}^{n} \sqrt{r}. \]

### Step 2: Change to Riemann Sum
We recognize that as \( n \) approaches infinity, the sum can be approximated by an integral. To do this, we can express \( \sqrt{r} \) in terms of \( n \):

Let \( x = \frac{r}{n} \), which implies \( r = nx \). The increment \( \Delta r \) can be approximated as \( n \Delta x \) where \( \Delta x = \frac{1}{n} \).

Thus, we can rewrite the sum:

\[ \sum_{r=1}^{n} \sqrt{r} = \sum_{r=1}^{n} \sqrt{n x} = \sqrt{n} \sum_{r=1}^{n} \sqrt{x} \Delta x. \]

### Step 3: Set Up the Integral
The limit now becomes:

\[ \lim_{n \to \infty} \frac{\sqrt{n}}{n \sqrt{n}} \sum_{r=1}^{n} \sqrt{x} \Delta x = \lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} \sqrt{x} \Delta x. \]

As \( n \to \infty \), this sum approaches the integral:

\[ \int_{0}^{1} \sqrt{x} \, dx. \]

### Step 4: Calculate the Integral
Now we compute the integral:

\[ \int_{0}^{1} \sqrt{x} \, dx. \]

The antiderivative of \( \sqrt{x} \) is:

\[ \frac{2}{3} x^{3/2}. \]

Evaluating this from 0 to 1 gives:

\[ \left[ \frac{2}{3} x^{3/2} \right]_{0}^{1} = \frac{2}{3}(1) - \frac{2}{3}(0) = \frac{2}{3}. \]

### Step 5: Final Result
Thus, the value of the limit is:

\[ \lim_{n \to \infty} \frac{\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n}}{n \sqrt{n}} = \frac{2}{3}. \]

### Conclusion
The final answer is:

\[ \frac{2}{3}. \]