SUMMARY
The discussion centers on the partial sum of the cosine telescoping series defined by the expression Ʃ (cos(1/(n)^2) - cos(1/(n+1)^2)). It is established that the nth partial sum results in cos(1) - cos(1/(n+1)^2). As n approaches infinity, this limit evaluates to cos(1) - 1, which is negative. The use of Wolfram Alpha confirms this result, demonstrating that partial sums can indeed be negative.
PREREQUISITES
- Understanding of telescoping series
- Familiarity with limits in calculus
- Basic knowledge of trigonometric functions, specifically cosine
- Experience with computational tools like Wolfram Alpha
NEXT STEPS
- Explore the properties of telescoping series in greater detail
- Study the behavior of limits involving trigonometric functions
- Learn how to use Wolfram Alpha for series and limits
- Investigate other examples of series that yield negative partial sums
USEFUL FOR
Students studying calculus, mathematicians interested in series convergence, and anyone exploring the properties of trigonometric functions in series.