SUMMARY
The series defined by the sum of (3^n + 4^n) / (3^n + 5^n) converges. By applying the comparison test, it is established that 3^n + 5^n is greater than 5^n, leading to the inequality 1/(3^n + 5^n) < 1/5^n. Consequently, the terms 3^n/(3^n + 5^n) and 4^n/(3^n + 5^n) can be bounded by (3/5)^n and (4/5)^n respectively, both of which are convergent geometric series.
PREREQUISITES
- Understanding of infinite series and convergence criteria
- Familiarity with the comparison test for series
- Knowledge of geometric series and their properties
- Basic algebraic manipulation of inequalities
NEXT STEPS
- Study the comparison test for convergence in more detail
- Learn about geometric series and their convergence criteria
- Explore advanced series convergence techniques such as the ratio test
- Practice solving similar series problems involving exponential functions
USEFUL FOR
Students studying calculus, particularly those focusing on series convergence, as well as educators looking for examples of applying the comparison test in mathematical proofs.