Calc 2 series quiz question

Question

Is [An] = (((2n^2)+(2n)+1)/(((n^2)+n)^2)), monotonic? then

Write in function form and find the derivative of f'(n) = (((2n^2)+(2n)+1)/(((n^2)+n)^2))

Determine where the sequence [An] = (natural log(n))/(square root(n)), is monotonic and give the GUB or LUB.

I have no clue what GUB or LUB is also this was all the information I was given in the problem

Answer 1

For the first sequence to determine if it is monotonic you want to determine if it is always increasing or always decreasing. You can do that by examining the derivative of the corresponding function:

f(x) = (2x²+2x+1)/(x²+x)²

Applying the quotient rule and simplifying leads to:

f'(x)= -(4x+2)(x²+x+1)/(x²+x)³

For x ≥ 1, the first derivative is always negative, which means the original function is always decreasing.

a₁ = 5/4 and the is going to be the greatest value the sequence ever achieves, So it is an upper bound and also a least upper bound.

The limit of a(n) as n->∞ is 0 (the degree of the denominator is greater than the degree of the numerator. So, zero is a lower bound. Since the sequence is bounded and monotonic it converges (to a limit of 0).

The 2nd sequence defined by g(x) = ln(x)/√x has a derivative (2-ln(x))/(2x^3/2). For x≥1, the 1st derivative is equal to zero at e²≈7.4, Since g'(2), for example, is positive and g'(8) is negative, the original function increases from 1 to e² and decreases after that. So the sequence is NOT monotonic. You can apply L'Hopitals rule to determine that the limit of the sequence is also zero.

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