Math 2.0

Approximations and expansions Proof

Proof of Taylor's theorem with the Peano remainder

Builds on Taylor's theoremTaylor polynomialLittle o notationLimit of a function

Proof

We rewrite the conclusion of Taylor's theorem in little o notation.

  1. Let Pk(x)=∑j=0kf(j)(a)j!(x−a)jP_k(x) = \sum_{j=0}^{k} \frac{f^{(j)}(a)}{j!} (x - a)^j, the Taylor polynomial of degree kk of ff at aa.
  2. As in little o notation, with x→ax \to a in place of x→∞x \to \infty, the claim means the following. For each ε>0\varepsilon > 0 there is δ>0\delta > 0 with ∣f(x)−Pk(x)∣≤ε∣x−a∣k|f(x) - P_k(x)| \le \varepsilon |x - a|^k for all x∈Ix \in I with ∣x−a∣<δ|x - a| < \delta.
  3. By Taylor's theorem, there is hk ⁣:I→ℝh_k \colon I \to \R with hk(x)→0h_k(x) \to 0 as x→ax \to a and f(x)−Pk(x)=hk(x)(x−a)kf(x) - P_k(x) = h_k(x)(x - a)^k for all x∈Ix \in I.
  4. Let ε>0\varepsilon > 0. By the definition of the limit applied to (−ε,ε)(-\varepsilon, \varepsilon), there is δ>0\delta > 0 with ∣hk(x)∣<ε|h_k(x)| < \varepsilon for x∈Ix \in I and 0<∣x−a∣<δ0 < |x - a| < \delta.
  5. For such xx, ∣f(x)−Pk(x)∣=∣hk(x)∣ ∣x−a∣k≤ε∣x−a∣k|f(x) - P_k(x)| = |h_k(x)| \, |x - a|^k \le \varepsilon |x - a|^k.
  6. For x=ax = a, step 3 gives f(a)−Pk(a)=0f(a) - P_k(a) = 0. So the inequality of step 2 holds at aa as well.
  7. Hence f(x)=Pk(x)+o((x−a)k)f(x) = P_k(x) + o\big((x - a)^k\big) as x→ax \to a.

Idea

Taylor's theorem already writes the error as hk(x)(x−a)kh_k(x)(x - a)^k with hk→0h_k \to 0. Saying that the error is o((x−a)k)o((x - a)^k) is just another way to say that this factor hkh_k tends to 00.

Reading path

To understand this page from the ground up, read these 158 pages in order. The foundations come first. The path takes the proofs Proof of Taylor's theorem by repeated use of L'Hôpital's rule, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the linearity of differentiation, Proof of the sequential characterization of limits, Proof of the Cauchy mean value theorem from Rolle's theorem, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that differentiable functions are continuous, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of L'Hôpital's rule from the Cauchy mean value theorem, Proof of the Archimedean property in an Archimedean ordered field, Proof that a composition of continuous functions is continuous, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the mean value theorem from Rolle's theorem, Proof of the constant multiple rule.

  1. Big O notation definition
  2. Construction of the reals from Cauchy sequences concept
  3. Definition by recursion concept
  4. Element definition
  5. Integers object
  6. Limit point of a set definition
  7. Local maximum definition
  8. Mathematical induction concept
  9. Natural numbers object
  10. Ordered field definition
  11. Peano axioms concept
  12. Proposition concept
  13. Rational numbers object
  14. Real numbers object
  15. Sequence definition
  16. Set bounded above definition
  17. Little o notation definition
  18. Set definition
  19. Ring of integers statement
  20. Limit of a function definition
  21. Local minimum definition
  22. Archimedean ordered field definition
  23. Successor function definition
  24. Conjunction definition
  25. Disjunction definition
  26. Implication definition
  27. Negation definition
  28. Predicate definition
  29. The reals form a field statement
  30. Real sequence definition
  31. Subsequence definition
  32. Infimum definition
  33. Supremum definition
  34. Ordered pair definition
  35. Continuity at a point definition
  36. Derivative definition
  37. Indeterminate form concept
  38. Addition of natural numbers definition
  39. Proof of the Archimedean property in an Archimedean ordered field proof
  40. The rationals form a field statement
  41. Biconditional definition
  42. The successor function is injective statement
  43. Zero is not a successor statement
  44. Truth table concept
  45. Modus ponens concept
  46. Universal quantifier definition
  47. Bounded sequence definition
  48. Limit of a sequence definition
  49. Monotone sequence definition
  50. Completeness axiom statement
  51. Cartesian product definition
  52. Continuous function definition
  53. Differentiable function definition
  54. Archimedean property statement
  55. Proof of the commutativity of addition by induction proof
  56. Order on the natural numbers definition
  57. Subset definition
  58. The rationals form an ordered field statement
  59. Propositional logic concept
  60. Mathematical proof concept
  61. Convergent sequence definition
  62. Sequential continuity definition
  63. Proof that a composition of continuous functions is continuous proof
  64. Derivative function definition
  65. Binary relation definition
  66. Order on the reals definition
  67. Commutativity of addition statement
  68. Proof of strong induction from ordinary induction proof
  69. Tautology definition
  70. Higher derivative definition
  71. Antisymmetric relation definition
  72. Reflexive relation definition
  73. Transitive relation definition
  74. Interval definition
  75. Strong induction statement
  76. Logical equivalence definition
  77. Taylor polynomial definition
  78. Preorder definition
  79. Proof of the epsilon-delta form of continuity proof
  80. Proof of the epsilon-delta form of the limit of a function proof
  81. Proof of the well-ordering principle from strong induction proof
  82. Proof of the double negation law by a truth table proof
  83. Partial order definition
  84. Epsilon-delta definition of continuity statement
  85. Epsilon-delta definition of a limit statement
  86. Well-ordering principle statement
  87. Double negation law statement
  88. Total order definition
  89. Composition of continuous functions is continuous statement
  90. Proof by contradiction concept
  91. Proof that every sequence has a monotone subsequence, by peaks proof
  92. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  93. Every sequence has a monotone subsequence statement
  94. The reals form an ordered field statement
  95. Absolute value definition
  96. Proof of the approximation property of the supremum proof
  97. Proof of the triangle inequality for real numbers proof
  98. Proof that the absolute value is multiplicative proof
  99. Proof of the epsilon-N definition of the limit of a sequence proof
  100. Approximation property of the supremum statement
  101. Triangle inequality statement
  102. Absolute value is multiplicative statement
  103. Epsilon-N definition of the limit of a sequence statement
  104. Proof that the reals are Archimedean from the completeness axiom proof
  105. Proof of the limit of a sum of sequences proof
  106. Proof of the squeeze theorem for sequences proof
  107. Proof of the uniqueness of limits proof
  108. Proof that differentiable functions are continuous proof
  109. Proof that limits preserve non-strict inequalities proof
  110. The reals are Archimedean statement
  111. Proof that convergent sequences are bounded proof
  112. Proof of the monotone convergence theorem for sequences proof
  113. Differentiable functions are continuous statement
  114. Proof that there is a natural number n with 1/n below any positive epsilon proof
  115. Limit of a sum of sequences statement
  116. Uniqueness of limits statement
  117. Limits preserve non-strict inequalities statement
  118. Squeeze theorem statement
  119. Convergent sequences are bounded statement
  120. Monotone convergence theorem for sequences statement
  121. Proof of the limit of a product of sequences proof
  122. There is a natural number n with 1/n below any positive epsilon statement
  123. Proof that a decreasing sequence bounded below converges proof
  124. Proof of the sequential characterization of limits proof
  125. Decreasing sequence bounded below converges statement
  126. Proof that continuity is equivalent to sequential continuity proof
  127. Limit of a product of sequences statement
  128. Continuity is equivalent to sequential continuity statement
  129. Sequential characterization of limits statement
  130. Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
  131. Proof of the chain rule with an auxiliary difference quotient proof
  132. Proof of the product rule proof
  133. Proof of the linearity of differentiation proof
  134. Bolzano–Weierstrass theorem statement
  135. Chain rule statement
  136. Product rule statement
  137. Linearity of differentiation statement
  138. Proof of the boundedness theorem by Bolzano–Weierstrass proof
  139. Proof of the power rule by induction proof
  140. Proof of the constant multiple rule proof
  141. Boundedness theorem statement
  142. Power rule statement
  143. Constant multiple rule statement
  144. Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
  145. Extreme value theorem statement
  146. Proof of Fermat's theorem on stationary points proof
  147. Fermat's theorem on stationary points statement
  148. Proof of Rolle's theorem proof
  149. Rolle's theorem statement
  150. Proof of the Cauchy mean value theorem from Rolle's theorem proof
  151. Proof of the mean value theorem from Rolle's theorem proof
  152. Mean value theorem statement
  153. Cauchy mean value theorem statement
  154. Proof of L'Hôpital's rule from the Cauchy mean value theorem proof
  155. L'Hôpital's rule statement
  156. Proof of Taylor's theorem by repeated use of L'Hôpital's rule proof
  157. Taylor's theorem statement
  158. Proof of Taylor's theorem with the Peano remainder proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Taylor%27s_theorem, Statement of the theorem
About this page
TagQYJMFCWDZB
URIm2://QYJMFCWDZB
Kindproof
WritingW0 stub
MSC41A58
ProvesTaylor's theorem with the Peano remainder
UsesTaylor's theorem, Taylor polynomial, Little o notation, Limit of a function
Changed4 November 2026

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