SUMMARY
The limit $$\lim_{n\to\infty} (-1)^n n^3 + 2^{-n}$$ does not exist due to the divergent nature of the alternating sequence of cubes, which oscillates between positive and negative values. While the term $$2^{-n}$$ approaches zero as $$n$$ approaches infinity, the dominant term $$(-1)^n n^3$$ diverges. Therefore, L'Hopital's rule is not applicable in this scenario, confirming that the limit is undefined.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with alternating sequences
- Knowledge of L'Hopital's rule and its applications
- Basic concepts of convergence and divergence in sequences
NEXT STEPS
- Study the properties of alternating sequences in calculus
- Learn about the application and limitations of L'Hopital's rule
- Explore the concept of divergence in sequences and series
- Investigate other methods for evaluating limits involving oscillating functions
USEFUL FOR
Students of calculus, mathematicians analyzing limits, and educators teaching concepts of convergence and divergence in sequences.