Evaluate : `lim_(n-> oo) (1^4+2^4+3^4+...+n^4)/n^5 - lim_(n->oo) (1^3+2^3+...+n^3)/n^5`

Options

A. `1/30`
B. zero
C. `1/4`
D. `1/5`

Answer

To evaluate the expression

\[ \lim_{n \to \infty} \frac{1^4 + 2^4 + 3^4 + \ldots + n^4}{n^5} - \lim_{n \to \infty} \frac{1^3 + 2^3 + \ldots + n^3}{n^5}, \]

we will use the standard formulas for the sums of powers of integers.

### Step 1: Evaluate the first limit

The sum of the first \( n \) natural numbers raised to the fourth power is given by the formula:

\[ \sum_{k=1}^{n} k^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30}. \]

Substituting this into the limit, we have:

\[ \lim_{n \to \infty} \frac{\sum_{k=1}^{n} k^4}{n^5} = \lim_{n \to \infty} \frac{\frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30}}{n^5}. \]

### Step 2: Simplify the first limit

This simplifies to:

\[ \lim_{n \to \infty} \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30n^5}. \]

Dividing the numerator and denominator by \( n^5 \):

\[ = \lim_{n \to \infty} \frac{(1)(1 + \frac{1}{n})(2 + \frac{1}{n})\left(3 + \frac{3}{n} - \frac{1}{n^2}\right)}{30}. \]

### Step 3: Evaluate the limit

As \( n \to \infty \), the terms \( \frac{1}{n} \) and \( \frac{1}{n^2} \) approach 0. Thus, we have:

\[ = \frac{1 \cdot 1 \cdot 2 \cdot 3}{30} = \frac{6}{30} = \frac{1}{5}. \]

### Step 4: Evaluate the second limit

The sum of the first \( n \) natural numbers raised to the third power is given by the formula:

\[ \sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2. \]

Substituting this into the limit, we have:

\[ \lim_{n \to \infty} \frac{\sum_{k=1}^{n} k^3}{n^5} = \lim_{n \to \infty} \frac{\left(\frac{n(n+1)}{2}\right)^2}{n^5}. \]

### Step 5: Simplify the second limit

This simplifies to:

\[ = \lim_{n \to \infty} \frac{n^2(n+1)^2}{4n^5} = \lim_{n \to \infty} \frac{(1)(1 + \frac{1}{n})^2}{4}. \]

### Step 6: Evaluate the limit

As \( n \to \infty \), the term \( \frac{1}{n} \) approaches 0. Thus, we have:

\[ = \frac{1 \cdot 1^2}{4} = \frac{1}{4}. \]

### Step 7: Combine the results

Now we can combine the results from the two limits:

\[ \frac{1}{5} - \frac{1}{4}. \]

To combine these fractions, we find a common denominator, which is 20:

\[ = \frac{4}{20} - \frac{5}{20} = -\frac{1}{20}. \]

### Final Answer

Thus, the final answer is:

\[ \boxed{-\frac{1}{20}}. \]