Increasing, decreasing or monotonic sequences

Question

Is the sequence increasing, decreasing, or monotonic?

a. {(1⋅3⋅5...(2n-1)) / n!}

b. {((-1)^n n^3) / (2n^3 + 1n^2 +1)}

c. {n^(e) e^(-n)}

(How do I show my work for this question)

Answer 1

You are clearly using different terminology then what I am used to.

So let me give you my definitions, and then answer your questions based on those.

Hopefully you will be able to translate my answers to your terminology.

Definition:

A sequence {x_n} is strictly increasing iff for each n, x_n < x_n+1.

A sequence {x_n} is strictly decreasing iff for each n, x_n > x_n+1.

A sequence {x_n} is monotone increasing iff for each n, x_n ≤ x_n+1.

A sequence {x_n} is monotone decreasing iff for each n, x_n ≥ x_n+1.

A sequence {x_n} is monotone iff it is monotone increasing or monotone decreasing.

Note that these five categories are not exclusive.

For example, a strictly increasing sequence is also monotone increasing.

And a sequence in any one of the first four categories is also monotone.

The statement of the problem does not say whether sequence elements starts with x₀ or x₁.

However, the way problem a. is written I will assume that sequences start with x₁.

There are several ways to solve these kinds of problems and show your work.

For example, you could express the general difference between any two neighboring

elements, and show that it is always positive, or always negative.

For your given examples, it turns out more advantageous to take the ration of

consecutive elements and compare it to 1.

a.

x_n = 1.3.5...(2n-1)/n!

x_n+1 = 1.3.5...(2(n+1)-1) / (n+1)! =

1.3.5...(2n+1) / (n!(n+1)) =

1.3.5...(2n-1)(2n+1) / (n!(n+1)) =

x_n (2n+1) / (n+1)

Thus

x_n+1/x_n = (2n+1) / (n+1) > 1 for any n > 0

Therefore the sequence is strictly increasing.

b.

The sequence is not monotone because even elements are positive and odd elements are negative.

c.

x_n = n^e e^-n

x_n+1 = (n+1)^e e^-(n+1)

Thus

x_n+1/x_n = (n+1)^e e^-(n+1) / n^e e^-n = ((n+1)/n)^e / e

For n=1

x₂/x₁ = 2^e/e > 1

On the other hand, for n sufficiently large (n+1)/n is very close to 1, and

1^e/e < 1.

For example, you can check that for n = 100.

Since the ration between successive elements is sometime < 1 and sometimes >1,

the sequence is not monotone.