Proving that a function is monotonic and bounded

In summary: Yeah right again I miscalculated the factor \frac{i+9}{2i-1}=\frac{1}{2}+\frac{1}{2} \frac{19}{2i-1}In summary, the conversation is about solving a series problem involving the ratio of terms and finding an upper and lower bound. The conversation also includes hints and corrections to calculations. The main tips given are to simplify the problem and to prove that the series is decreasing for large values of n.
  • #1
transgalactic
1,395
0
in this link i written the question and how i tried to solve them

http://img504.imageshack.us/img504/7371/95405842kw4.gif

how to finish it??
 
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  • #2
I think you may be making the monotone part a bit too complicated. Look at [tex]x_{n+1}/x_n[/tex]. What can you tell about this ratio?

For the bounded part, why try to look at an upper bound for the numerator and a lower bound for the denominator?
 
  • #3
Your series is [tex]x_n=\prod_{i=1}^n \frac{i+9}{2i-1} [/tex]. I'll give you a hint, [tex]\frac{i+9}{2i-1}=\frac{1}{4} \frac{19}{2i-1}[/tex]. What happens as i increases (after i=5)?

Good luck
 
  • #4
why i need to prove that An+1<An
??
 
  • #5
If you can show that A_(n+1)<A_n for n greater than some value, then it's decreasing for large enough n - which means its monotonic.
 
  • #6
Focus said:
Your series is [tex]x_n=\prod_{i=1}^n \frac{i+9}{2i-1} [/tex]. I'll give you a hint, [tex]\frac{i+9}{2i-1}=\frac{1}{4} \frac{19}{2i-1}[/tex].
Is there a typo in the line above? I don't see how this could be true.
Focus said:
What happens as i increases (after i=5)?

Good luck
 
  • #7
Mark44 said:
Is there a typo in the line above? I don't see how this could be true.

Yeah sorry that should be[tex] \frac{i+9}{2i-1}=\frac{1}{2}+\frac{1}{4} \frac{19}{2i-1}[/tex]
 
  • #8
Focus said:
Yeah sorry that should be[tex] \frac{i+9}{2i-1}=\frac{1}{2}+\frac{1}{4} \frac{19}{2i-1}[/tex]
I don't see that those two expressions are equal, either. On the right side you get 4i - 17 in the numerator, and 4(2i - 1) in the denominator.
 
  • #9
Mark44 said:
I don't see that those two expressions are equal, either. On the right side you get 4i - 17 in the numerator, and 4(2i - 1) in the denominator.

Yeah right again I miscalculated the factor [tex]\frac{i+9}{2i-1}=\frac{1}{2}+\frac{1}{2} \frac{19}{2i-1}[/tex]
 

FAQ: Proving that a function is monotonic and bounded

How do you prove that a function is monotonic?

To prove that a function is monotonic, you need to show that its derivative is either always positive or always negative. This means that the function is either always increasing or always decreasing.

What is the importance of proving monotonicity in a function?

Proving monotonicity is important because it helps us understand the behavior of a function. It tells us whether the function is consistently increasing or decreasing, and this information can be useful in making predictions and analyzing data.

What does it mean for a function to be bounded?

A function is bounded if its values do not get infinitely large or infinitely small. In other words, the function is limited and does not have extreme values that approach infinity or negative infinity.

How can you prove that a function is bounded?

To prove that a function is bounded, you need to show that its values fall within a certain range. This can be done by finding the maximum and minimum values of the function, or by using mathematical techniques such as the squeeze theorem.

Can a function be both monotonic and bounded?

Yes, a function can be both monotonic and bounded. For example, a linear function is both monotonic (always increasing or decreasing) and bounded (limited by its y-intercept and slope).

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