SUMMARY
The limit of the geometric sequence defined by the expression $$\lim_{{n}\to{\infty}} {(\frac{2}{3})}^{n}$$ evaluates to 0. This conclusion is reached by recognizing that as n approaches infinity, the term $$\left(\frac{2}{3}\right)^n$$ can be rewritten using the exponential function as $$e^{n\ln(2/3)}$$. Since $$\ln(2/3)$$ is negative, the limit simplifies to $$\lim_{n\to\infty} e^{-n\ln(3/2)}$$, which confirms that the limit approaches 0.
PREREQUISITES
- Understanding of geometric sequences
- Familiarity with limits in calculus
- Knowledge of exponential functions and logarithms
- Basic proficiency in mathematical notation and manipulation
NEXT STEPS
- Study the properties of geometric sequences and their limits
- Learn about the behavior of exponential functions as their exponents approach infinity
- Explore the concept of logarithmic functions and their applications in limits
- Investigate other types of sequences and their convergence criteria
USEFUL FOR
Students in calculus, mathematics educators, and anyone interested in understanding limits of sequences and series.