SUMMARY
The sum of the series \( \sum_{n=1}^\infty \frac{n}{(n+1)!} \) converges to 1. The sequence of partial sums is given by \( S_1 = \frac{1}{2} \), \( S_2 = \frac{5}{6} \), \( S_3 = \frac{23}{24} \), \( S_4 = \frac{119}{120} \), and \( S_5 = \frac{719}{720} \). By observing the pattern in the partial sums, one can conjecture that \( S_n = 1 - \frac{1}{(n+1)!} \). This can be proven using mathematical induction, and taking the limit as \( n \) approaches infinity confirms that the sum equals 1.
PREREQUISITES
- Understanding of infinite series and convergence
- Familiarity with factorial notation and properties
- Basic knowledge of mathematical induction
- Ability to compute limits in calculus
NEXT STEPS
- Study the concept of convergence in infinite series
- Learn about mathematical induction techniques
- Explore the properties of factorials and their applications in series
- Investigate other series summation techniques, such as the ratio test
USEFUL FOR
Mathematics students, educators, and anyone interested in series convergence and mathematical proofs.