Math 2.0

Sequences, series, summability Proof

Proof of term-by-term differentiation

Builds on Series of functionsLinearity of differentiationOne-sided derivativeSum of a seriesUniform convergence of derivatives allows term-by-term differentiation

Proof

We apply the theorem on sequences of differentiable functions to the partial sums.

  1. Let a<ba < b, and let fnf_n and x0x_0 be as in the statement. Let sN=∑n=0Nfns_N = \sum_{n=0}^{N} f_n be the partial sums of the series ∑fn\sum f_n.
  2. Each sNs_N is differentiable on [a,b][a, b], with sN′=∑n=0Nfn′s_N' = \sum_{n=0}^{N} f_n'. This follows from linearity of differentiation and induction on NN.
  3. At the endpoints, the derivatives are one-sided. The proof of linearity applies to one-sided limits in the same way, since a sum of difference quotients is the difference quotient of the sum.
  4. By hypothesis, ∑nfn(x0)\sum_n f_n(x_0) converges. So the sequence (sN(x0))(s_N(x_0)) converges, by the definition of a convergent series.
  5. By hypothesis, ∑nfn′\sum_n f_n' converges uniformly on [a,b][a, b]. By step 2, this means that (sN′)(s_N') converges uniformly on [a,b][a, b] to the function g=∑n=0∞fn′g = \sum_{n=0}^{\infty} f_n'.
  6. By steps 2, 4 and 5, the theorem on uniform convergence of derivatives applies to (sN)(s_N).
  7. It gives that (sN)(s_N) converges uniformly on [a,b][a, b] to a differentiable function ss, with s′=gs' = g.
  8. By the definition of a series of functions, step 7 says that ∑fn\sum f_n converges uniformly on [a,b][a, b] with sum ss. So the sum is differentiable, and (∑n=0∞fn)′=s′=g=∑n=0∞fn′.\Big( \sum_{n=0}^{\infty} f_n \Big)' = s' = g = \sum_{n=0}^{\infty} f_n'.

Idea

A series of functions is the sequence of its partial sums. Finite sums can be differentiated term by term. So the result for sequences, applied to the partial sums, gives the result for series.

Reading path

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

Where this leads

437 pages build on this one.

Further reading

  1. Web https://dec41.user.srcf.net/notes/IB_M/analysis_ii_thm_proof.pdf, Section 1, Uniform convergence
About this page
Tag4JTEKAZY93
URIm2://4JTEKAZY93
Kindproof
WritingW0 stub
MSC40A30, 26A24
ProvesTerm-by-term differentiation
UsesSeries of functions, Linearity of differentiation, One-sided derivative, Sum of a series, Uniform convergence of derivatives allows term-by-term differentiation
Changed4 November 2026

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