How do you formally show subsequent element in a series is larger?

In summary, the conversation discusses how to formally show that the inequality s_{n+1}>\frac{s^{2}_{n}+1}{2}>s_{n} is true, using mathematical induction and the fact that the square of a non-zero real number is always positive. The conversation also explores whether it is acceptable to simply state that fact as a proof.
  • #1
golmschenk
36
0
As an example, for the sequence:
[tex]s_{1} =2[/tex]
[tex]s_{n+1}=\frac{s^{2}_{n}+1}{2}[/tex]
We see:
When [tex]s_{n} >1 [/tex] then,
[tex]s_{n+1}=\frac{s^{2}_{n}+1}{2}>s_{n}[/tex]
But how do I formally show that this last inequality is true? This example is fine or just one similar to it. Thanks!

EDIT: Many of those superscripts are suppose to be subscripts. I don't seem to be able to edit them so there correct though. Sorry, hopefully you can still figure it out. If, it doesn't have any subscript at all, then whatever is the superscript is suppose to be the subscript. However, if there is a subscript then the superscript is correct.
 
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  • #2
If s_n > 1, is (s_n-1)^2>0?
 
  • #3
Yes, but I'm not quite sure how to use that to formally explain that the inequality is true. Also, again, I know that what you asked is true, but how do you formally show this?
 
  • #4
You might try showing this by mathematical induction. Clearly, S2 > S1, so that establishes your base case.

Assume that Sn > Sn - 1, and see if you can use this assumption and your formula for Sn to show that Sn + 1 > Sn.
 
  • #5
golmschenk said:
Yes, but I'm not quite sure how to use that to formally explain that the inequality is true. Also, again, I know that what you asked is true, but how do you formally show this?
You're not sure how to show that the square of a non-zero number is greater than zero?
 
  • #6
zhentil said:
You're not sure how to show that the square of a non-zero number is greater than zero?

Not formally, no. And that's what this thread was about to begin with. Is it ok just to state that the square of a non-zero, real number is positive?

Mark44 said:
You might try showing this by mathematical induction. Clearly, S2 > S1, so that establishes your base case.

Assume that Sn > Sn - 1, and see if you can use this assumption and your formula for Sn to show that Sn + 1 > Sn.

Thanks, I think I've got it mostly figured out.
 
  • #7
golmschenk said:
Not formally, no. And that's what this thread was about to begin with. Is it ok just to state that the square of a non-zero, real number is positive?
Surely, for any real x, [itex]x^2\ge 0[/itex] which means "either [itex]x^2= 0[/itex] or [itex]x^2> 0[/itex]. Further, if [itex]x^2= 0[/itex] then, taking the square root of both sides, x= 0. Therefore, if [itex]x\ne 0[/itex], [itex]x^2> 0[/itex].
That's a "formal" proof.

Thanks, I think I've got it mostly figured out.
 

1. How do you formally show that subsequent elements in a series are larger?

To formally show that subsequent elements in a series are larger, you can use mathematical notation such as n+1 > n to represent that the next element in the series is greater than the current element. Graphs or tables can also be used to visually demonstrate the increase in values as the series progresses.

2. What is the significance of showing that subsequent elements in a series are larger?

Showing that subsequent elements in a series are larger helps to establish a pattern or trend in the data. This can be useful in making predictions or drawing conclusions about the series. It also provides evidence for the validity of the series and its potential usefulness in further analysis.

3. Are there any alternative ways to show that subsequent elements in a series are larger?

Yes, there are alternative ways to show that subsequent elements in a series are larger. One method is to use comparative language such as "greater than", "larger than", or "more than" to describe the relationship between the elements. Another way is to use percentage or ratio comparisons to show the increase in values.

4. How do you handle outliers or anomalies when showing that subsequent elements in a series are larger?

When showing that subsequent elements in a series are larger, outliers or anomalies should be examined and potentially removed from the data if they are determined to be erroneous. If the outlier is found to be a valid data point, it should be included and clearly labeled in the analysis. In some cases, it may be necessary to use a logarithmic scale to better visualize the trend in the series.

5. Can subsequent elements in a series be smaller or equal to the previous element?

Yes, subsequent elements in a series can be smaller or equal to the previous element. It is not uncommon for a series to fluctuate or have variations in values, so it is important to analyze the data carefully and consider the context of the series. However, if the series is intended to show a clear increase in values, it is important to accurately and consistently represent that in the data.

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