High School How to Develop in Series: Definition & Examples

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Developing in series involves using a Taylor expansion to approximate functions, such as the relativistic energy of particles. The third term in the expansion does not have a specific name or significance. For high velocities, the relativistic energy formula should be applied, while the Newtonian approximation suffices at lower velocities. Understanding these distinctions is crucial for accurate energy measurements in particle physics. Accurate application of these concepts ensures proper analysis of particle behavior at varying speeds.
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How to develop in series as in the image below? What does the third term signify?
1462258388878.jpg
 
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It is a simple Taylor expansion of the relativistic energy.
 
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Thank you.
 
What does the third term signify?
1462261373421.jpg
 
It doesn't have a name or any particular meaning as far as I am aware.

If you are measuring the energy of particles, say, when v is high enough for this term to be measurable then you ought to be using the relativistic formula for energy. Below that velocity the Newtonian approximation is good enough.
 
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Thread 'Am I understanding the concept of proper frame of reference?'

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