Question

why does S sub n = 8(1/2 - 1/(n+2))?

http://calcchat.tdlc.com/solutionart/etf5e/09/b/se09b01043.png

Answer 1

Because when you add the terms of Sn, all the middle terms cancel each other out leaving only the first and the last: 1/2 - (1/(n+2))

Answer 2

Look at the attached diagram. When you expand the term in increasing \(n\) you see that you obtain a term \[-\frac{1}{3} +\frac{1}{3}\] Illustrated in the red circles. This continues through out the expression as \(n\) tends to infinity, meaningthat the middle terms cancel. Note that the last two terms will be

\[-\frac{1}{n+1}+\frac{1}{n+1}+\frac{1}{n+2}\] where the first two tems in this expression cancel. Overall the only two terms that don't cancel are the \(1/2\) and the \(\frac{1}{n+2}\) terms leaving only \[\frac{1}{2}+\frac{1}{n+2}\]