Which condition is required for a function f(x) to actually equal its Taylor series on an interval, not just be approximated by it?
Af must be continuous on the interval
Bf must be infinitely differentiable on the interval
CThe remainder term R_n(x) must approach zero as n → ∞
DThe series must converge for all real numbers, not just on a finite interval
Infinite differentiability is necessary but not sufficient. The Taylor series of f always converges to *some* value, but that value may not equal f(x) unless the remainder Rn(x) = f(x) - (partial sum of n terms) → 0 as n → ∞. The classic counterexample is f(x) = e^(-1/x²) at x = 0: all derivatives equal 0 there, so the Taylor series is identically 0, which does not equal f(x) for x ≠ 0.
Question 2 True / False
A Taylor polynomial and a Taylor series for the same function centered at the same point are two names for the same mathematical object.
TTrue
FFalse
Answer: False
A Taylor polynomial is a finite sum of n+1 terms — it approximates f(x) near the center point. A Taylor series is an infinite sum. Within the radius of convergence, the Taylor series (when it equals f) gives exact values; the polynomial always retains error. The distinction matters: polynomials are used for computation and estimates; the series is used when exact representation is needed.
Question 3 Short Answer
The Taylor series for e^x is 1 + x + x²/2! + x³/3! + ···, and it converges for all real x. What does it mean to say this series 'converges to e^x'?
Think about your answer, then reveal below.
Model answer: It means that as you add more and more terms of the series, the partial sums get arbitrarily close to the exact value of e^x — and in the limit, the infinite sum equals e^x exactly, not just approximately.
Convergence means the sequence of partial sums S_n = 1 + x + x²/2! + ··· + xⁿ/n! approaches e^x as n → ∞. This is stronger than mere approximation: for any desired level of accuracy, you can find an n large enough that S_n is within that accuracy of e^x. For e^x this holds for every real number x.