The Maclaurin Series — A Clean Derivation

Many smooth functions can be written as an infinite polynomial. When this expansion is centred at x = 0, we obtain the Maclaurin series. This article derives the Maclaurin formula directly from repeated differentiation, showing precisely why the coefficients involve derivatives and factorials.

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1) Begin with a General Power Series

Suppose a function f(x) can be expressed as

f(x) = a₀ + a₁x + a₂x² + a₃x³ + … + aᵣxʳ + …

The constants aᵣ are real coefficients whose values we wish to determine.

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2) Evaluate at x = 0

f(0) = a₀

so

a₀ = f(0).

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3) Differentiate Once

f′(x) = a₁ + 2a₂x + 3a₃x² + … + r·aᵣxʳ⁻¹ + …

Setting x = 0 eliminates all higher powers:

f′(0) = a₁.

Thus,

a₁ = f′(0).

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4) Differentiate Again

f″(x) = 2·1·a₂ + 3·2·a₃x + … + r(r−1)aᵣxʳ⁻² + …

Evaluate at x = 0:

f″(0) = 2! · a₂

Hence

a₂ = f″(0) / 2!.

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5) The General Pattern

Differentiate repeatedly. After r differentiations, all terms still containing x vanish at x = 0, leaving only the coefficient from the term aᵣxʳ:

f(r)(0) = r! · aᵣ.

Therefore,

aᵣ = f(r)(0) / r!.

This is the key identity. The factorial appears naturally because each differentiation multiplies by a descending integer.

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6) Assemble the Maclaurin Series

Substitute these coefficients back into our original power series:

f(x) = f(0)
     + f′(0)x
     + f″(0)x² / 2!
     + f‴(0)x³ / 3!
     + …
     + f(r)(0)xʳ / r!
     + …

In sigma notation:

f(x) = Σ ( f(r)(0) / r! ) xʳ,

where the sum runs from r = 0 to infinity.

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Why This Works

Differentiation isolates coefficients. Each time we differentiate, we bring down an integer from the exponent. After r differentiations, the term aᵣxʳ becomes r! · aᵣ, while all other terms still contain a factor of x that vanishes at x = 0. Thus, only r! · aᵣ survives, making

aᵣ = f(r)(0) / r!.

This explains both the derivatives and the factorials in the Maclaurin series.

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Example: The Exponential Function

For f(x) = eˣ, every derivative is also eˣ. Evaluating at x = 0 gives

f(0) = f′(0) = f″(0) = … = 1.

So

eˣ = 1 + x + x²/2! + x³/3! + …

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Summary

• Write f(x) as a power series.
• Differentiate repeatedly.
• Evaluate each derivative at x = 0.
• Each coefficient is aᵣ = f⁽ʳ⁾(0) / r!.
• Substitute these expressions back into the series.

This derivation shows that the Maclaurin series is not a mysterious formula; it is simply the natural outcome of repeated differentiation and evaluation at the origin. Once seen through this lens, the structure becomes direct and elegant.

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Notes for personal study.