Prove convergence of a series of real data

[AlexSaly]

Hello,

 

I have obtained data from experiments, and when I plot the values I obtained vs the time, I can see that at large t, it's going towards something constant.

 

I would like to prove that it converges (or find a criterion to determine "from this t, we can already see that the data will converge, so we can stop the measurement at time t"... something like that ).

 

I have never tried to do a "convergence proof" with real data, so I am a bit lost...

 

 

Thanks for your help!

[studiot]

Do you have a mathematical model of whatever this data is about?

[AlexSaly]

Thanks for replying.

 

No, I don't have a model. I think I know how I would proceed from a model, but I was wondering first if there existed some criterion to show convergence directly from the data... Do you think there is a way to do that?

[studiot]

I don't know what sort of proof you are looking for. You have provided scant data to work with (pun intended).

 

Let us say that your data, when plotted, comprises a series of increasing ordinates, but increasing at an apparently ever decreasing rate.

 

If you can find a convergent mathematical sequence, each term of which, produces an ordinate larger than corresponding ordinates in your data, then you have a mathematical proof that your data is convergent, over the range taken.

 

You should look up the comparison test and ratio test for series/sequences.

 

However be warned these can produce an infinite range of data, yours cannot so the conclusion to infinity would be extrapolation. This may not be the case in reality.

 

For instance in reality measuring the output of an electrical power supply has the above characteristic. As you increase the load on the supply the output slowly 'droops' until it suddenly falls to zero as the supply protection kicks in.

[AlexSaly]

Okay, got it Thank you very much for your help!

[studiot]

If in doubt - ask.

 

[AriePiron]

Electrical engineers use Fourier transforms quite often to determine the relationships between parts of a circuit. Solving a relationship directly in the time domain would be too complicated, but by transforming the time equations into the frequency domain, the frequency response can easily be calculated using algebra. Once that happens, the result can undergo an inverse Fourier transform to yield a useful answer in the time domain.

The series of functions can be put into the form

 

∑k=0∞fk(x)

Edited October 5, 2013 by AriePiron