AP® Calculus BC · Unit 10 · Topic 10.15
Representing Functions as a Power Series
Any function that looks like 1/(1 - something) can be written as a geometric power series by matching that something to the ratio r. The series equals the function only inside its radius of convergence.
The geometric series formula says 1/(1-r) = 1 + r + r2 + r3 + ... whenever |r| < 1. Read that backwards: whenever you see a function built from 1/(1 - something), that something is just r in disguise, and the function already is an infinite sum. 1/(1-x) is the plainest case: r = x, so 1/(1-x) = 1 + x + x2 + x3 + ..., valid for |x| < 1.
The real move is substitution. Replace x in the standard series with whatever expression sits in your function's denominator slot, then simplify. For 1/(1+3x), write it as 1/(1-(-3x)), so r = -3x, giving the series 1 - 3x + 9x2 - 27x3 + ..., converging when |-3x| < 1, meaning |x| < 1/3. For x/(1-x2), start from 1/(1-x2) with r = x2, giving 1 + x2 + x4 + ..., then multiply everything by x to get x + x3 + x5 + ..., convergent for |x| < 1. The extra factor of x out front doesn't touch convergence.
Outside the radius of convergence, the series and the function part ways: the series diverges while the function itself might still be perfectly well-defined. The equality is only a local promise. This matters right away, because once a function is written as a power series, you can differentiate or integrate that series term by term to build power series for related functions like ln(1+x) or arctan(x), and the radius of convergence carries over almost unchanged.
Key terms
- power series
- an infinite sum of the form sum of cn (x-a)n, built from increasing powers of x.
- geometric series
- a series of the form sum of rn, which converges to 1/(1-r) exactly when |r| < 1.
- radius of convergence
- the distance from the center within which a power series converges to its function.
- substitution
- replacing the variable r in a known series with a new expression to build a series for a related function.
Watch it explained
- How it worksHow does substituting a new expression for x reshape the geometric series into a new function?
- Worked exampleFind a power series for 1/(1 + 3x) and state where it converges.
- Worked exampleFind a power series for x/(1 - x2) starting from the geometric series.
- Common mistakeWhy doesn't the geometric series representation still hold outside its radius?
- Ties togetherHow does representing this function as a power series set up differentiating and integrating that same series term by term next?
Check yourself
End of topic.