Some Algebra Problems

Date: 6/1/96 at 21:56:38
From: Anonymous
Subject: Series

Conjecture but do not prove the formula for 
   1/2.3 + 1/3.4 + 1/4.5 +...+ n. Show all work.

Date: 6/2/96 at 7:37:40
From: Doctor Anthony
Subject: Re: Series

The sum of this series is easily found using the method of 
differences.

Note first that 1/2.3 = 1/2 - 1/3
                1/3.4 = 1/3 - 1/4
                1/4.5 = 1/4 - 1/5
                .................
                .................
             1/n(n+1) = 1/n - 1/(n+1)  
         1/(n+1)(n+2) = 1/(n+1) - 1/(n+2)
        ----------------------------------

Add all the equations. The lefthand side gives the series we want, 
and on the righthand side we note that terms cancel between lines, 
i.e. -1/3 in the first line cancels with +1/3 in the second line. In 
fact every term cancels except the first term on the first line and 
the last term on the last line.

Sum of series = 1/2 - 1/(n+2)

We note from this that the sum to infinity would be 1/2. If we add the 
two terms we get sum of series = n/{2(n+2)}

-Doctor Anthony,  The Math Forum
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