Study the monotony of the string xn = 1/(n-1), n>=2.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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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