`lim_(nrarroo) ((1^(k)+2^(k)+3^(k)+"......"n^(k)))/((1^(2)+2^(2)+"....."+n^(2))(1^(3)+2^(3)+"....."+n^(3))) = F(k)`, then `(k in N)`

Options

A. `F(k)` is finite for`k le 6 (B)`
B. `F(5) = 0`
C. `F(6) = 12/7`
D. `F(6) = 5/7`

Answer

To solve the limit

\[ \lim_{n \to \infty} \frac{1^k + 2^k + 3^k + \ldots + n^k}{(1^2 + 2^2 + \ldots + n^2)(1^3 + 2^3 + \ldots + n^3)} = F(k), \]

we will analyze the numerator and denominator separately.

### Step 1: Analyze the numerator

The numerator is the sum of the first \( n \) natural numbers raised to the power \( k \). We can express this sum using the formula:

\[ \sum_{i=1}^{n} i^k \sim \frac{n^{k+1}}{k+1} \quad \text{as } n \to \infty. \]