SUMMARY
The infinite series 1 - 2^{-1/2} + 3^{-1/2} - 4^{-1/2} + 5^{-1/2} + ... can be expressed in LaTeX as \(\sum_{n=1}^\infty\frac{(-1)^{n-1}}{\sqrt{n}}\). This series converges as it qualifies as an alternating Leibniz series, where the general term sequence decreases monotonically to zero while alternating in sign. The convergence of this series is confirmed by the properties of alternating series.
PREREQUISITES
- Understanding of LaTeX typesetting
- Familiarity with infinite series and convergence criteria
- Knowledge of alternating series, specifically the Leibniz test
- Basic mathematical notation and functions, including square roots
NEXT STEPS
- Study the properties of alternating series and the Leibniz test for convergence
- Learn how to typeset mathematical expressions using LaTeX
- Explore the Dirichlet eta function and its applications
- Investigate other types of convergent series and their characteristics
USEFUL FOR
Mathematicians, students of calculus, educators teaching series convergence, and anyone interested in LaTeX for mathematical documentation.