Question

In this problem we evaluate the series:

Answer 1

Answer:

  A.  1/(n(n+2)) = (1/2)/n - (1/2)/(n+2)

  B.  14651/19800

Step-by-step explanation:

A.

The coefficients of the partial-fraction expansion can be found from ...

  f(n) = 1/(n(n+2)) = A/n +B/(n+2)

  n·f(x) = 1/(n+2) = A +Bn/(n+2)

For n=0, this becomes ...

  1/(0 +2) = A = 1/2

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Similarly, ...

  (n+2)·f(n) = 1/n = A(n+2)/n +B

For n = -2, this becomes ...

  1/(-2) = B = -1/2

The n-th term can be written ...

  1/(n(n+2)) = (1/2)/n - (1/2)/(n+2)

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B.

The sum is ...

  1/(1·3) +1/(2·4) +1/(3·5) +... +1/(98·100)

  = 1/2(1/1 -1/3 +1/2 -1/4 +1/3 -1/5 +... +1/98 -1/100)

  = 1/2((1/1 +1/2 +1/3 +...1/98) -(1/3 +1/4 +1/5 +...+1/100)

We notice that terms 3..98 cancel, so the sum is ...

  = 1/2(1/1 +1/2 -1/99 -1/100) = (1/2)(3/2 -199/9900)

  = 14651/19800