A fraction can be a polynomial that never ends.

In Year 1 the binomial expansion stopped. For negative and fractional powers it runs on forever, and the picture shows both how well it works and exactly where it breaks.

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This is $y = (1 + x)^{-1}$. The binomial expansion with $n = -1$ turns it into an endless polynomial:

Add the terms one at a time. Each new polynomial follows the curve further, but only between $x = -1$ and $x = 1$. Outside that, the polynomials peel away, up and down in turn.

The terms are powers of $x$: they shrink only when $|x| < 1$. At $x = 2$ the sums go $1, -1, 3, -5, \dots$ and never settle, although $(1 + 2)^{-1} = \tfrac{1}{3}$. Always state where an expansion is valid.

Near $x = 0$ the higher powers are tiny, so a few terms make a good approximation. At $x = 0.1$ four terms give $1 - 0.1 + 0.01 - 0.001 = 0.909$, and the true value is $\tfrac{1}{1.1} = 0.9090\ldots$

Any power, as long as x is small enough.

Square roots, $(a + bx)^n$, approximations such as $\sqrt{2}$, series through partial fractions, and twelve graded exam questions with full worked solutions are waiting.

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