Problem #PRU-60427

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

Problem

Find the sums of the following series:

a) \({\frac {1} {1 \times 2}} + {\frac {1} {2 \times 3}} + {\frac {1} {3 \times 4}} + {\frac {1} {4 \times 5}} + \dots\);

b) \({\frac {1} {1 \times 2 \times 3}} + {\frac {1} {2 \times 3 \times 4}} + {\frac {1} {3 \times 4 \times 5}} + {\frac {1} {4 \times 5 \times 6}} + \dots\);

c) \({\frac {0!} {r!}} + {\frac {1!} {(r-1)!}} + {\frac {2!} {(r-2) !}} + {\frac {3!} {(r-3)!}} + \dots\) for \(r \geq 2\).