Study the monotony of the string xn = 1/(n-1), n>=2.
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We have xn = 1 / (n-1) and we have to study the monotony for n>=2.
To find the monotony of xn for n>=2, we find the derivative of xn with respect to n
(xn)' = -1/ (n-1)^2
for n>=2, xn' = -1 / (2 - 1)^2 < 0.
Therefore xn is decreasing for all value of n >=2.
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To determine if the string, whose general term is xn = 1/(n-1) is increasing or decreasing, we'll have to determine if the difference between 2 consecutive terms of the string is positive or negative.
We'll have to determine xn+1 = 1/(n+1-1)
xn+1 = 1/n
Now, we'll calculate the difference:
xn+1 - xn = 1/n - 1/(n-1)
xn+1 - xn = (n-1-n)/n(n-1)
xn+1 - xn = -1/n(n-1)
Since n>=2, the result of the difference is negative:
-1/n(n-1) < 0 => xn+1 - xn < 0 => xn+1 < xn
The string is strictly decreasing: xn+1 < xn.
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