`Lim_(n->oo)[1/n^2 * sec^2 (1/n^2)+2/n^2 * sec^2 (4/n^2)+..............+1/n * sec^2 1]`

Explanation

To solve the limit problem given, we will break it down step by step.

### Step 1: Rewrite the Limit Expression
We start with the limit expression:
\[ \lim_{n \to \infty} \left[ \frac{1}{n^2} \sec^2\left(\frac{1}{n^2}\right) + \frac{2}{n^2} \sec^2\left(\frac{4}{n^2}\right) + \ldots + \frac{1}{n} \sec^2(1) \right] \]
This can be rewritten as:
\[ \lim_{n \to \infty} \sum_{r=1}^{n} \frac{r}{n^2} \sec^2\left(\frac{r^2}{n^2}\right) \]

### Step 2: Recognize the Riemann Sum
As \( n \to \infty \), the expression resembles a Riemann sum for the integral:
\[ \int_0^1 x \sec^2(x^2) \, dx \]

### Step 3: Change of Variables
To evaluate the integral, we will perform a substitution. Let:
\[ t = x^2 \implies dt = 2x \, dx \implies dx = \frac{dt}{2\sqrt{t}} \]
The limits change as follows:
- When \( x = 0 \), \( t = 0 \)
- When \( x = 1 \), \( t = 1 \)

### Step 4: Substitute in the Integral
Substituting into the integral:
\[ \int_0^1 x \sec^2(x^2) \, dx = \int_0^1 \frac{1}{2} \sec^2(t) \, dt \]

### Step 5: Evaluate the Integral
The integral of \( \sec^2(t) \) is:
\[ \int \sec^2(t) \, dt = \tan(t) \]
Thus:
\[ \int_0^1 \sec^2(t) \, dt = \tan(1) - \tan(0) = \tan(1) - 0 = \tan(1) \]
Therefore:
\[ \int_0^1 x \sec^2(x^2) \, dx = \frac{1}{2} \tan(1) \]

### Step 6: Final Result
Putting it all together, we find:
\[ \lim_{n \to \infty} \left[ \frac{1}{n^2} \sec^2\left(\frac{1}{n^2}\right) + \frac{2}{n^2} \sec^2\left(\frac{4}{n^2}\right) + \ldots + \frac{1}{n} \sec^2(1) \right] = \frac{1}{2} \tan(1) \]

### Final Answer
\[ \frac{1}{2} \tan(1) \]