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Posted by Soroban on September 13, 2002 at 13:34:07:

In Reply to: Please write your problem correctly.... posted by Subhotosh Khan on September 13, 2002 at 12:39:18:

: : Is 1/n(n+2) a telescoping series, if so, how do I convert this to 1/n-n/n+c form?
: *****************************************

: In general, you will have to solve like this:
: =A/n + B/(n+2)
: =[A*(n+2) + B*n]/[n*(n+2)]

: so
: A*(n+2) + B*n = 1 ....from here you get
: n(A+B) = 0

: so A + B = 0 .....(1)
: AND
: 2*A = 1 ...........(2)

: Solve (1) and (2) to get the proper values.

: However, this problem can be solved very easily by "observation" (which may not be possible all the time).....

Good point, Mr. K!

Have you or Habaneroeater seen this method for Partial Fractions?

Begin as you did: 1/n(n+2) = A/n + B/(n+2)

Clear denominators: 1 = A(n+2) + Bn

Let n = 0, and we get immediately: A = 1/2
Let n = -2, and we get: B = -1/2

You see, we select n so that one (or more) terms is/are zero, and solve.

This method is dependable and very fast.

However, there are "purists" who will not allow or use this method.

This is completely understandable. Examine the orignal fraction.
We find that 0 and -2 are not in the domain of n.
Yet we use the very values that are outlawed.

A moral dilemma: use the shorter "impure" method or do it the long way
and remain chaste and pure?

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