Topic 10.15 Representing Functions as Power Series: manipulate known power series by substitution, multiplication, differentiation and integration to represent new functions and evaluate otherwise intractable integrals (BC).

What this topic is asking

The College Board (Topic 10.15, BC only) is the capstone of Unit 10: using the standard power series as building blocks and manipulating them to represent new functions and to evaluate integrals that have no elementary antiderivative. This is where series stop being abstract and become a practical computational tool.

The manipulation toolkit

Why integrating a series beats hunting for an antiderivative

The deepest use of this topic is integrating functions with no elementary antiderivative. Functions like $e^{-x^2}$, $\frac{\sin x}{x}$, and $\frac{1}{1 + x^4}$ cannot be antidifferentiated in closed form, so the Fundamental Theorem is no direct help. But each has a simple power series, and a power series can always be integrated term by term, producing a series for the integral. For $\int_0^1 e^{-x^2}\,dx$, you write $e^{-x^2} = 1 - x^2 + \frac{x^4}{2!} - \cdots$, integrate each term to get $x - \frac{x^3}{3} + \frac{x^5}{5\cdot 2!} - \cdots$, and evaluate. The result is a convergent series whose partial sums approximate the integral to any desired accuracy, often an alternating series, so the error is bounded by the first omitted term (Topic 10.10). This is the practical reason BC develops series at all.

A worked integral via series

A worked function representation

Combining series and the interval of validity

You can also add, subtract, and multiply series to represent combinations of functions, lining up like powers of $x$. When you manipulate a series, the radius of convergence is preserved by substitution (adjusted for the substituted variable), differentiation, and integration, though the endpoints can gain or lose convergence. For example, integrating $\frac{1}{1+x} = \sum(-1)^n x^n$ (valid on $|x| < 1$) gives the series for $\ln(1 + x)$, which now also converges at the endpoint $x = 1$ (giving the alternating harmonic series for $\ln 2$). Always state the interval on which your representation is valid, and remember that the operations of this topic are only legitimate inside the interval of convergence.