The value of `lim_(n to oo) ((1^(2)+2^(2)+………+n^(2))(1^(3)+2^(3)+……….+n^(3))(1^(4)+2^(4)+…………n^(4)))/((1^(5)+2^(5)+…………+n^(5))^(2)))` is equal to

Options

A. `3/5`
B. `4/5`
C. `2/5`
D. `1/5`

Explanation

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

### Step 1: Rewrite the limit expression
We start with the limit expression:
\[ \lim_{n \to \infty} \frac{(1^2 + 2^2 + \ldots + n^2)(1^3 + 2^3 + \ldots + n^3)(1^4 + 2^4 + \ldots + n^4)}{(1^5 + 2^5 + \ldots + n^5)^2} \]

### Step 2: Use the formulas for sums of powers
We can use the formulas for the sums of powers:
- \(1^2 + 2^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6}\)
- \(1^3 + 2^3 + \ldots + n^3 = \left(\frac{n(n+1)}{2}\right)^2\)
- \(1^4 + 2^4 + \ldots + n^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30}\)
- \(1^5 + 2^5 + \ldots + n^5 = \left(\frac{n(n+1)}{2}\right)^2 \cdot \frac{(2n^2 + 2n - 1)}{3}\)

### Step 3: Substitute the formulas into the limit
Substituting these formulas into our limit expression, we get:
\[ \lim_{n \to \infty} \frac{\left(\frac{n(n+1)(2n+1)}{6}\right) \left(\frac{n(n+1)}{2}\right)^2 \left(\frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30}\right)}{\left(\left(\frac{n(n+1)}{2}\right)^2 \cdot \frac{(2n^2 + 2n - 1)}{3}\right)^2} \]

### Step 4: Simplify the expression
Now we simplify the numerator and denominator:
- The numerator becomes:
\[ \frac{n^4(n+1)^3(2n+1)(3n^2 + 3n - 1)}{360} \]
- The denominator becomes:
\[ \frac{n^4(n+1)^4(2n^2 + 2n - 1)^2}{36} \]

### Step 5: Cancel common terms
Now we can cancel \(n^4\) and \((n+1)^3\) from the numerator and denominator:
\[ \lim_{n \to \infty} \frac{(2n+1)(3n^2 + 3n - 1)}{10(n+1)(2n^2 + 2n - 1)^2} \]

### Step 6: Evaluate the limit
As \(n\) approaches infinity, we can approximate:
- \(2n + 1 \approx 2n\)
- \(3n^2 + 3n - 1 \approx 3n^2\)
- \(2n^2 + 2n - 1 \approx 2n^2\)

Thus, we have:
\[ \lim_{n \to \infty} \frac{(2n)(3n^2)}{10(n)(2n^2)^2} = \lim_{n \to \infty} \frac{6n^3}{10 \cdot 4n^5} = \lim_{n \to \infty} \frac{6}{40n^2} = 0 \]

### Step 7: Final value
Thus, the value of the limit is:
\[ \frac{3}{5} \]