Question

The value of \[ \lim_{n \to \infty} \left(3^n + 5^n + 7^n \right)^{\tfrac{1}{n}} \] is _________.

Hint:

When taking limits of the form \((a_1^n + a_2^n + \dots + a_k^n)^{1/n}\), the largest base among \(a_i\) dominates as \(n \to \infty.\)

Answer 1

Correct Answer: 7

Step 1: Identify dominant term.
For large \(n\), among \(3^n, 5^n, 7^n,\) the term \(7^n\) dominates the sum.
Step 2: Simplify the limit.
\[ \lim_{n \to \infty} \left(3^n + 5^n + 7^n\right)^{1/n} = \lim_{n \to \infty} 7 \left(\left(\frac{3}{7}\right)^n + \left(\frac{5}{7}\right)^n + 1\right)^{1/n}. \]
Step 3: Evaluate limit inside parentheses.
As \(n \to \infty,\) \(\left(\frac{3}{7}\right)^n, \left(\frac{5}{7}\right)^n \to 0.\) Thus, \[ \left(\left(\frac{3}{7}\right)^n + \left(\frac{5}{7}\right)^n + 1\right)^{1/n} \to 1. \]
Step 4: Conclusion.
\[ \lim_{n \to \infty} (3^n + 5^n + 7^n)^{1/n} = 7. \]