Taylor Series Fundamentals and Applications

This question is built around Taylor and Maclaurin expansions in one variable, with a focus on how the remainder term works and what it actually tells you about approximation error. The setup starts from the familiar power series for standard functions and the finite Taylor polynomial with its remainder in Lagrange form. Candidates are then asked to think carefully about the role of the mysterious intermediate point in that remainder term, and to use the machinery to prove a basic inequality involving integer powers, valid on a specified range of the variable. The second part is a classic inequality that often appears in calculus-based proofs and analysis arguments.

Solving it leans heavily on understanding the Lagrange form of the Taylor remainder, bounding derivatives on an interval, and exploiting the sign of a derivative to control the sign of the error. The inequality component invites either a Taylor expansion argument or an equivalent calculus approach using convexity and second derivatives. An interviewer is watching for comfort with switching between series, derivatives, and inequalities, as well as precision about where intermediate points live and how bounds are justified rigorously.

What it tests

Taylor series expansions are local polynomial approximations of differentiable functions, constructed using the function's derivatives at a point. The general structure is that each term in the series captures increasingly subtle information about the function's behavior near the expansion point, with the remainder term quantifying the error of truncating the series. The power of this approach is that it translates the global behavior of a function into a sum of local, algebraic pieces, making analysis and approximation tractable. The reason this works is that differentiability ensures the function can be closely matched by its tangent (and higher-order) polynomials in a neighborhood, and the remainder term formalizes how quickly the approximation improves as more terms are included. The pattern is that, for analytic functions, the Taylor series converges to the function within a certain radius, and the error can be tightly bounded using the next derivative.