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Coefficients

Level 1 - Math I (Physics) topic page in Power Series.

Principle

The coefficients of a power series determine the size and sign of each power. They control both the function represented by the series and how quickly the terms shrink.

Power series coefficients
\[f(x)=a_0+a_1(x-c)+a_2(x-c)^2+\cdots\]

When a function has a Taylor series about \(c\), its coefficients come from derivatives at the centre. The constant term gives the value at the centre, and higher coefficients measure local rates of change.

Notation

\(a_n\)
coefficient of (x-c)^n
\(f^{(n)}(c)\)
nth derivative of f evaluated at c
\(n!\)
factorial n(n-1)\cdots2\cdot1, with 0!=1
\(a_0\)
constant term, usually the value at the centre
\(a_1\)
linear coefficient, usually the first derivative at the centre

Coefficients can be read directly from a series that is already written in powers of \((x-c)\). If the powers are not in that form, rewrite first.

Method

Step 1: Match powers with the same centre

Put the expression in ascending powers of \((x-c)\):

Coefficient matching form
\[a_0+a_1(x-c)+a_2(x-c)^2+\cdots\]

Then compare the coefficient beside each power.

Step 2: Use derivative data for Taylor coefficients

If the series is a Taylor series for \(f\) about \(c\), use

Taylor coefficient
\[a_n=\frac{f^{(n)}(c)}{n!}\]

The first few cases show the pattern:

Value term
\[a_0=f(c)\]
Slope term
\[a_1=f'(c)\]
Quadratic term
\[a_2=\frac{f''(c)}{2!}\]
Cubic term
\[a_3=\frac{f'''(c)}{3!}\]

Step 3: Check whether the coefficients shrink fast enough

Large coefficients can make terms fail to shrink. Convergence tests compare \(a_{n+1}(x-c)^{n+1}\) with \(a_n(x-c)^n\) to see whether the terms become small.

Rules

Taylor coefficient formula
\[a_n=\frac{f^{(n)}(c)}{n!}\]
Coefficient extraction
\[a_n=\text{coefficient of }(x-c)^n\]
Constant term
\[a_0=f(c)\]
Linear term
\[a_1=f'(c)\]

Examples

Question
Find
\[a_0,a_1,a_2\]
in
\[4+7(x-2)-3(x-2)^2\]
Answer
The powers are already centred at
\[c=2\]
Match coefficients:
\[a_0=4\]
\[a_1=7\]
and
\[a_2=-3\]

Checks

  • Coefficients must be matched using the same centre.
  • The coefficient of \((x-c)^2\) is not automatically the second derivative; it is the second derivative divided by \(2!\).
  • A zero coefficient means that power is absent.
  • Coefficients can be physical constants when the series models measured behaviour.

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Radius of Convergence