10.15 Representing Functions as Power Series

Manipulating the geometric series

Start from 1/(1-x) = Σxⁿ for |x|<1. Substitute expressions for x (like -x, x², or 2x) to build power series for related functions, such as 1/(1+x) = Σ(-1)ⁿxⁿ or 1/(1+x²) = Σ(-1)ⁿx²ⁿ. The interval of convergence shifts based on the substitution, so re-check it.

Term-by-term differentiation & integration

Within its interval of convergence, a power series can be differentiated or integrated term by term, producing a new power series with the same radius of convergence (endpoints may change). This lets you find series for functions like ln(1+x) or arctan(x) by integrating known series for 1/(1+x) or 1/(1+x²).

Building new series from known ones

Combine substitution, multiplication by xⁿ, differentiation, and integration to represent new functions as power series centered at a point. Always verify the interval of convergence using the ratio test after transforming the series, since operations like integration can add convergence at an endpoint.

Key terms

Geometric series formula

1/(1-x) = Σxⁿ for |x| < 1; the foundation for building other power series by substitution.

Series for 1/(1+x)

Σ(-1)ⁿxⁿ, found by substituting -x into 1/(1-x) = Σxⁿ.

Term-by-term differentiation/integration

A power series can be differentiated or integrated term by term within its interval of convergence, keeping the same radius of convergence.

Series for ln(1+x)

Obtained by integrating the series for 1/(1+x) term by term: x - x²/2 + x³/3 - ...

Series for arctan(x)

Obtained by integrating the series for 1/(1+x²): x - x³/3 + x⁵/5 - ...

Try an AP-style question

Which is a power series for 1/(1 + x)?

1. Σ (−1)ⁿ xⁿ/n
2. Σ xⁿ
3. Σ xⁿ/n!
4. Σ (−1)ⁿ xⁿ