Finding k for Gamma Function Convergence

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SUMMARY

The discussion focuses on determining the values of k for which the limit of the Gamma function, specifically \(\lim_{x \to \infty} \frac{\Gamma(kx + 1)}{x^{kx}}\), converges. It is established that for k < 0, the function diverges due to the properties of the Gamma function. Additionally, it is confirmed that for k > e, divergence occurs as well, utilizing Stirling's approximation. The convergence for k = 0 is also noted, while further analysis is required for the interval (0, e].

PREREQUISITES
  • Understanding of the Gamma function and its properties
  • Familiarity with limits and convergence in calculus
  • Knowledge of Stirling's approximation
  • Basic concepts of asymptotic analysis
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  • Research the properties of the Gamma function in detail
  • Study Stirling's approximation and its applications in asymptotic analysis
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bomba923
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For what values of k would
\mathop {\lim }\limits_{x \to \infty } \frac{{\Gamma \left( {kx + 1} \right)}}<br /> {{x^{kx} }}
converge?
 
Last edited:
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Certainly for k < 0 it diverges, since gamma is ill-behaved there. For k > e it also diverges by Stirling's approximation. There's the easy part! I'll have to think about the remaining (0, e]. (It obviously converges for k = 0.)
 
Last edited:

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