Question

Evaluate the limit: $$ \lim_{n \to \infty} \frac{17^7 + 27^7 + \dots + n^{77}}{n^{78}}. $$ 

• A. \( \frac{1}{77} \)
• B. \( 1 \)
• C. \( 76 \)
• D. \( \frac{1}{78} \)

Hint:

For limits involving summations of power functions, use the formula: \[ \sum_{k=1}^{n} k^m \approx \frac{n^{m+1}}{m+1} \] for large \( n \).

Answer 1

The Correct Option is D

Step 1: Understanding the given sum
The given sum in the numerator is: \[ S = 17^7 + 27^7 + \dots + n^{77}. \] The number of terms in the sum is approximately \( n \), and the dominant term in the summation is: \[ \sum_{k=1}^{n} k^{77}. \] Using the standard asymptotic sum formula: \[ \sum_{k=1}^{n} k^m \approx \frac{n^{m+1}}{m+1}, \] for large \( n \), we approximate: \[ \sum_{k=1}^{n} k^{77} \approx \frac{n^{78}}{78}. \] Step 2: Evaluating the limit
Substituting the approximation: \[ \lim_{n \to \infty} \frac{\sum_{k=1}^{n} k^{77}}{n^{78}} = \lim_{n \to \infty} \frac{\frac{n^{78}}{78}}{n^{78}}. \] \[ = \frac{1}{78}. \] Step 3: Conclusion
Thus, the correct answer is: \[ \frac{1}{78}. \]

Answer 2

To evaluate the limit \( \lim_{n \to \infty} \frac{17^7 + 27^7 + \dots + n^{77}}{n^{78}} \), we start by analyzing the series in the numerator. Consider the general term \( k^{77} \) where \( k \) ranges from 17 to \( n \). The numerator can be rewritten as a sum:

\( \sum_{k=17}^{n} k^{77} \).

The key is to approximate this sum using integral calculus for large \( n \). The sum \( \sum_{k=17}^{n} k^{77} \) can be closely approximated by the integral:

\( \int_{17}^{n} x^{77} \, dx \).

Calculating the integral, we have:

\( \int x^{77} \, dx = \frac{x^{78}}{78} + C \).

Evaluating at bounds:

\( \int_{17}^{n} x^{77} \, dx = \left[\frac{x^{78}}{78}\right]_{17}^{n} = \frac{n^{78}}{78} - \frac{17^{78}}{78} \).

The expression for the limit becomes:

\( \lim_{n \to \infty} \frac{\frac{n^{78}}{78} - \frac{17^{78}}{78}}{n^{78}} = \lim_{n \to \infty} \left(\frac{1}{78} - \frac{17^{78}}{78n^{78}}\right) \).

As \( n \to \infty \), the term \( \frac{17^{78}}{78n^{78}} \to 0 \). Hence, the limit simplifies to:

\( \frac{1}{78} \).

Therefore, the value of the limit is \( \frac{1}{78} \).