Taylor Series vs Taylor Polynomial
- is a representation of a function as a sum of terms that are calculated from the values of the function’s derivatives at a single point
- Taylor series is an infinite sum
- Taylor polynomial is a finite sum
- goal is to take a non-polynomial function and represent it as a polynomial function (which is easier to understand and manipulate)
- transforms: derivative information at a point → approximation information near that point
Taylor Series/Polynomial - Formula
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power series form |
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compact sigma form |
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where:
- 𝑛! denotes the factorial of 𝑛
- 𝑓(𝑛)(𝑎) denotes the 𝑛𝑡ℎ derivative of 𝑓 evaluated at the point 𝑎
Taylor Series/Polynomial - Formula Derivation
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We know that the power series can be defined as
- 𝑓(𝑥) = 𝛴0≤𝑛≤∞[𝑎𝑛𝑥𝑛] = 𝑎0 + 𝑎1𝑥 + 𝑎2𝑥2 + 𝑎3𝑥3+ …
When 𝑥 = 0,
- 𝑓(𝑥) = 𝑎0
So, differentiate the given function, it becomes,
- 𝑓’(𝑥) = 𝑎1 + 2𝑎2𝑥 + 3𝑎3𝑥2 + 4𝑎4𝑥3 +….
Again, when you substitute 𝑥 = 0, we get
- 𝑓‘(0) = 𝑎1
Therefore, [𝑓‘(0)/1!] = 𝑎1
So, differentiate it again, we get
- 𝑓”(𝑥) = 2𝑎2 + 6𝑎3𝑥 + 12𝑎4𝑥2 + …
Now, substitute 𝑥=0 in second-order differentiation, and we get
- 𝑓”(0) = 2𝑎2
Therefore, [𝑓”(0)/2!] = 𝑎2
By generalizing the equation, we get
- 𝑓𝑛(0)/𝑛! = 𝑎𝑛
Now substitute the values in the power series we get,
- 𝑓(𝑥) = 𝑓(0) + 𝑓‘(0)𝑥 + [𝑓”(0)/2!]𝑥2+ [𝑓'''(0)/3!]𝑥3+ ….
Generalize 𝑓 in a more general form, and it becomes
- 𝑓(𝑥) = 𝑏 + 𝑏1(𝑥-𝑎) + 𝑏2(𝑥-𝑎)2 + 𝑏3(𝑥-𝑎)3+ ….
Now, 𝑥 = 𝑎, we get
- 𝑏𝑛 = 𝑓𝑛(𝑎)/𝑛!
Now, substitute 𝑏n in a generalized form
- 𝑓(𝑥) = 𝑓(𝑎) + [𝑓’(𝑎)/1!](𝑥−𝑎) + [𝑓”(𝑎)/2!](𝑥−𝑎)2+ [𝑓'''(𝑎)/3!](𝑥−𝑎)3+ ….
Hence, the Taylor series is proved
Taylor Series/Polynomial - Variants
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Types |
Description |
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a Taylor series is centered at zero (i.e. a = 0) |