Number 1,3,5,6 please
2 answers:
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Answer:
A. 1/(n(n+2)) = (1/2)/n - (1/2)/(n+2)
B. 14651/19800
Step-by-step explanation:
<h3>A.</h3>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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<h3>B.</h3>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
