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It can be proved that $(1+1/n)^n$ is increasing $n\in \mathbb{N}$.

Now look at the picture given below, the dark $>$ signs are actually "greater" sign, and you can check that those inequality holds perfectly!

Now take a look at at the rectangle(there those arrows are vectors). Let us define vectors on $\vec{AB}$, if the value in $A$ is greater than the value in $B$. We denote $A_{ij} =(1+1/i)^j$. $"+" \text{and}\space "="$ defined as follows: $\vec{AB}+\vec{BC}=\vec{AC}$ is value of $A>$ value in $B>$ value in $C$, implies value of $A>$ value in $C$.

Consequently we also have $A_{nn}<A_{(n+1)(n+1)}$. This completes the whole thing.(ignore the word "Consequently")

On the other way we can also say that to preserve the system $A_{nn}<A_{(n+1)(n+1)}$ has to happen.

Which one is true??

The image is here

MAN-MADE
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  • You say: "Consequently we also have $A_{nn}<A_{(n+1)(n+1)}$. This completes the whole thing." What are you saying? That the picture leads to a proof of this fact? Or are you using that fact to draw the picture in the first place? – user49640 Jun 16 '17 at 08:27
  • @user49640 The word Consequently is wrong actually(Lack of experience of writing English), I was saying we also have $A_{nn}<A_{(n+1)(n+1)}$ – MAN-MADE Jun 16 '17 at 09:09
  • What do you mean by "This completes the whole thing"? – user49640 Jun 16 '17 at 09:53
  • @user49640 I mean that since we can prove that inequality the triangle property of vectors holds in every case and My question: Is this matrix happens to preserve "$>$" rule I explained or to preserve that rule $A_{nn}<A_{(n+1)(n+1)}$ is happening in this matrix? – MAN-MADE Jun 16 '17 at 10:04
  • Using induction may be instructive. – Jacob Wakem Jun 16 '17 at 15:56

4 Answers4

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A way of understanding intuitively the fact that the sequence is increasing is to note that it corresponds to componding (the same yearly) interest with greater and greater frequency. Compounding each week will yield higher profit than compounding each month, compounding each day will yield higher profit than compounding each week, etc. A nice "formal" proof appears here.

Mikhail Katz
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  • Can that intuition be converted into a proof? I mean, I don't know if this seems intuitive to me simply because it is something I know is true from hearing it said, or if I would have thought it true before hearing that it was true. It's not clear if you're saying that that proof is related to the intuition you mentioned. – user49640 Jun 16 '17 at 09:05
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Here is an easy way to see that your argument is flawed:

You could use the same argument to 'proof' that $\left(1+\frac{1}{n^2}\right)^n$ is increasing, but it is not.

MooS
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Your diagram of inequalities is inconclusive: for each square you only have proved \begin{matrix}A_{n,n+1}&<&A_{n,n}&<&A_{n+1,n}&\text{up-left corner}\\A_{n,n+1}&<&A_{n+1,n+1}&<&A_{n+1,n}&\text{down-right corner}\\ A_{n,n+1}&&&<&A_{n+1,n}&\text{diagonal SW-NE (redundant info, btw)} \end{matrix}

You are lacking the crucial information: the diagonal NW-SE. The "Consequently" is therefore not "consequently" at all. Basically, what you made is a diagram containing all the easy eastimates hat were inconclusive in the initial attempts of solution and, not surprisingly, they still don't work.

DanielWainfleet
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  • The word Consequently is wrong actually(Lack of writing English), I was saying we also have $A_{nn}<A_{(n+1)(n+1)}$ – MAN-MADE Jun 16 '17 at 09:08
  • @MANMAID And what I am saying is that it is true (it is in fact the thesis), but you have not proved it. –  Jun 16 '17 at 09:13
  • But it has been asked a lot of times in https://math.stackexchange.com , that's why I did not write the proof! Otherwise my post is also marked as duplicate! – MAN-MADE Jun 16 '17 at 09:18
  • Then, what's the question? I thought you were asking whether you had provided a sound proof. –  Jun 16 '17 at 09:20
  • My question: Is this matrix happens to preserve "$>$" rule I explained or to preserve that rule $A_{nn}<A_{(n+1)(n+1)}$ is happenning in this matrix? – MAN-MADE Jun 16 '17 at 09:30
  • My edit was for a minor typo: The word "no" in the 2nd-last sentence should have been "not". – DanielWainfleet Jun 16 '17 at 15:40
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You have five arrows in each square, and you seem to be saying that there is only one way to draw the sixth one. That is wrong. Consider the following two diagrams: $$\begin{matrix} 2 & 1 \\ 4 & 3 \end{matrix} \qquad \text{and} \qquad \begin{matrix} 3 & 1 \\ 4 & 2 \end{matrix} . $$

The five arrows that you have in your squares all go in the same direction in the two examples. In the first example, the sixth arrow goes in the direction you would like. But in the second example, it goes in the opposite direction to the one you would like.

user49640
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  • one can write anything using "$>$" rule I describe, but I am talking about this particular matrix. – MAN-MADE Jun 16 '17 at 09:50
  • You said "To preserve the system $A_{n,n} < A_{n+1,n+1}$ has to happen." I am showing why this is wrong. You can have two different systems in each square. If you are talking about this particular matrix, then please explain the difference between "happens in this particular matrix" and "has to happen in this particular matrix." – user49640 Jun 16 '17 at 09:55