Extension: Infinite Telescoping Series as a Limit
Read this topicProblems 3
- Let S=∑n=0∞cot−1(n2+n+1)S = \sum_{n=0}^{\infty} \cot^{-1}(n^2+n+1)S=∑n=0∞cot−1(n2+n+1). Find π/S\pi/Sπ/S. Source: HMMT February 2001
- Compute: ∑n=1∞1n(n+1).\sum_{n=1}^{\infty} \frac{1}{n(n+1)}.n=1∑∞n(n+1)1.
- ∑x=1∞1x∗(x+1)=12+16+112+…\sum_{x=1}^{\infty} \frac{1}{x*(x+1)} = \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+ \dotsx=1∑∞x∗(x+1)1=21+61+121+… Is this sum a finite value?