Taylor Slider - Taylor Series Approximation Practice

Sum a Taylor polynomial by hand - order 2, 3, or 4 for eˣ, sin, cos, or ln(1+x) at a given point - and see how fast the error shrinks.

Each round picks one of four functions - eˣ, sin(x), cos(x), or ln(1+x) - a sample point x, and an order from 2, 3, or 4 (kept low enough to sum by hand). You compute the Taylor (Maclaurin, centered at 0) approximation of that order at that x and type the number. There is no live readout - you find out only after committing.

The sample points are chosen per function: for ln(1+x) they all lie inside the series' interval of convergence (-1 < x ≤ 1), such as -0.9, -0.6, -0.4, 0.5, 0.8, 0.95; the other functions use points like -1.8, -1.3, -0.8, 0.6, 1.1, 1.5, 1.9. Each round also carries an error threshold of 0.1, 0.01, or 0.001 used in the reveal.

After checking, the reveal shows the exact order-n sum, the true function value, the resulting error, and whether your assigned order clears the round's error threshold - including the smallest order (up to 8) that does, or a note that even order 8 cannot reach it at that x.

Why quant interviews test this

Taylor expansion is arguably the most-used tool in quant interviews: quick numeric estimates (e^0.1, ln(1.05)), the log-return approximation ln(1+x) ≈ x - x²/2 behind volatility drag, Itô's lemma as a second-order expansion, delta-gamma P&L approximations, and duration-convexity in rates are all order-2 Taylor arguments. Being able to sum to order 3 or 4 by hand, cleanly, is table stakes.

The error question is the differentiator: interviewers follow up with how big is the error and when does the expansion fail. The next-dropped-term estimate and the convergence-interval caveat for ln(1+x) - the exact two facts this game's reveal drills - are the expected answers.

The Taylor Slider guide covers how scoring works, the strategy that wins, a worked example and the mistakes most players make.

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• Every game guide