SUMMARY
The series $\sum_{n=0}^{\infty}\frac{2^n}{3^nn!}$ converges to $e^{2/3}$. This conclusion is derived from recognizing that the series can be rewritten as $\sum_{n=0}^{\infty}(\frac{2}{3})^n \frac{1}{n!}$, which matches the standard exponential series formula $\sum_{n=0}^{\infty}\frac{x^n}{n!}=e^x$ with $x$ set to $\frac{2}{3}$. Therefore, the evaluation of the series is confirmed to be correct.
PREREQUISITES
- Understanding of infinite series and convergence
- Familiarity with the exponential function and its series expansion
- Basic knowledge of factorial notation and operations
- Ability to manipulate algebraic expressions involving series
NEXT STEPS
- Study the properties of exponential functions and their series expansions
- Explore convergence tests for infinite series
- Learn about Taylor series and their applications
- Investigate the relationship between series and differential equations
USEFUL FOR
Mathematicians, students studying calculus, and anyone interested in series convergence and exponential functions.