Show that the sequence is decreasing.

[maeveoneill]

There is an example in my calculus textbook that goes as follows:

Show tha the sequence an= n/(n^2 +1) is decreasing

Solution: We must show that an+1 < an, that is,
n + 1/ [(n +1)^2 +1) < n/ n^2 +1

The inequality is equivalent to the one we get by cross-multiplication:

n + 1/ [(n +1)^2 +1) < n/ n^2 +1 <=> (n+1)(n^2 +1) <n[(n+1)^2 +1]
. <=> n^3 + n^2 +n + 1 < n^3 +2n^2 +2n
. <=> 1 < n^2 +n

Since n > or equal 1, we know that the inequality n^2 +n >1 is true. Therefore an+1 <an and so {an} is decreasing.

I don't understand how this proves that the sequence is decreasing.. or why you would use the cross multiplication method. could you not just prove it using examples of direct substition.. anywyas if someone could explain how this means that the function is decreasing.. especially with regards to the last statement that would be wonderful. Thank youu..

 

[pka]

Looking at the sequence it is fairly easy to see that it is decreasing.
\(\displaystyle \begin{array}{l} a_n = \frac{n}{{n^2 + 1}} \\ a_1 = \frac{1}{2},\;a_2 = \frac{2}{5},\;a_3 = \frac{3}{{10}} \\ \end{array}\).

Decreasing means \(\displaystyle \left( {\forall n} \right)\left[ {a_{n + 1} < a_n } \right]\).
That is all that your textbook tried to show.

 

[daon]

could you not just prove it using examples of direct substition..

Is this your first math class using proofs? Ever hear of the quote "'For example...' is not a proof?"

Your text wanted to show \(\displaystyle a_n\) is always decreasing from one entry in the sequence to the next. When you give an example, for instance \(\displaystyle a_1 > a_2 > ... > a_{100}\) you are only showing that \(\displaystyle a_n\) is decreasing as n varies from 1 to 100. It could very well be true that \(\displaystyle a_{100} < a_{101}\). Thats why a proof is needed for all n such that \(\displaystyle a_n\) is defined, in this case 1 to infinity\(\displaystyle .\\

\\ \\\\\)

 

[tkhunny]

I find the algebra entirely unsatisfying. Anytime I see that profanity "cross multiply" I get nervous.

Since you are in Calculus, how about this:

\(\displaystyle \frac{d}{dx}\left(\frac{x}{x^{2}+1}\right) = \frac{1-x^{2}}{(x^{2}+1)^{2}}\)

For x > 1, it's pretty easy to argue that this guy is negative. What does that mean?