Math 2.0

Functions of a complex variable Proof

Proof that the binomial series has radius of convergence 1

Builds on Binomial seriesRatio formula for the radius of convergenceThere is a natural number n with 1/n below any positive epsilonEpsilon-N definition of the limit of a sequenceLimit of a sum of sequencesLimit of a product of sequences

Proof

We apply the ratio formula for the radius of convergence.

  1. Let α\alpha be a real number that is not a natural number. For each natural number kk, let ck=(αk)=α(α−1)⋯(α−k+1)/k!c_k = \binom{\alpha}{k} = \alpha (\alpha - 1) \cdots (\alpha - k + 1) / k!, the coefficients of the binomial series.
  2. For each natural number jj, α−j≠0\alpha - j \ne 0, since α\alpha is not a natural number. So ckc_k is a product of nonzero numbers divided by k!k!, and ck≠0c_k \ne 0 for all kk.
  3. From the formula in step 1, ck+1=ck⋅(α−k)/(k+1)c_{k+1} = c_k \cdot (\alpha - k)/(k + 1). So ∣ck+1∣/∣ck∣=∣α−k∣/(k+1)\abs{c_{k+1}} / \abs{c_k} = \abs{\alpha - k} / (k + 1).
  4. For k>∣α∣k > \abs{\alpha} we have α−k<0\alpha - k < 0. So ∣α−k∣/(k+1)=(k−α)/(k+1)=1−(1+α)/(k+1)\abs{\alpha - k}/(k + 1) = (k - \alpha)/(k + 1) = 1 - (1 + \alpha)/(k + 1).
  5. We show 1/(k+1)→01/(k + 1) \to 0. Let ε>0\varepsilon > 0. There is a natural number n≥1n \ge 1 with 1/n<ε1/n < \varepsilon.
  6. For k≥nk \ge n, 0<1/(k+1)<1/n<ε0 < 1/(k + 1) < 1/n < \varepsilon. By the epsilon-N form, 1/(k+1)→01/(k + 1) \to 0.
  7. By the product rule and the sum rule for limits, 1−(1+α)/(k+1)→1−0=11 - (1 + \alpha)/(k + 1) \to 1 - 0 = 1.
  8. By steps 3 and 4, ∣ck+1∣/∣ck∣→1\abs{c_{k+1}} / \abs{c_k} \to 1 as k→∞k \to \infty.
  9. By step 2 and step 8, the ratio formula applies with L=1L = 1. So the radius of convergence is R=1/1=1R = 1/1 = 1.

Idea

The ratio of consecutive coefficients is (α−k)/(k+1)(\alpha - k)/(k + 1), which tends to −1-1. So its absolute value tends to 11. The hypothesis on α\alpha makes every coefficient nonzero, so the ratio formula applies. If α\alpha were a natural number, the series would stop and be a polynomial.

Reading path

To understand this page from the ground up, read these 110 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof of the sum of a finite geometric series, Proof that there is a natural number n with 1/n below any positive epsilon, 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 limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of Pascal's rule from the recursive definition, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the comparison test, Proof that limits preserve non-strict inequalities, Proof of the factorial formula for binomial coefficients by induction, Proof of the binomial theorem by induction, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof of the ratio formula for the radius of convergence, Proof that a series of nonnegative terms converges iff its partial sums are bounded.

  1. Binomial coefficient definition
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Factorial definition
  5. Group concept
  6. Integers object
  7. Mathematical induction concept
  8. Ordered field definition
  9. Proposition concept
  10. Rational numbers object
  11. Real numbers object
  12. Sequence definition
  13. Set bounded above definition
  14. Proof of Pascal's rule from the recursive definition proof
  15. Set definition
  16. Ring definition
  17. Ring of integers statement
  18. Archimedean ordered field definition
  19. Conjunction definition
  20. Disjunction definition
  21. Implication definition
  22. Negation definition
  23. Predicate definition
  24. The reals form a field statement
  25. Real sequence definition
  26. Supremum definition
  27. Pascal's rule statement
  28. Ordered pair definition
  29. Commutative ring definition
  30. Biconditional definition
  31. The rationals form a field statement
  32. Truth table concept
  33. Modus ponens concept
  34. Universal quantifier definition
  35. Bounded sequence definition
  36. Geometric sequence definition
  37. Limit of a sequence definition
  38. Monotone sequence definition
  39. Series definition
  40. Completeness axiom statement
  41. Cartesian product definition
  42. Proof of the factorial formula for binomial coefficients by induction proof
  43. Subset definition
  44. The rationals form an ordered field statement
  45. Mathematical proof concept
  46. Propositional logic concept
  47. Proof of the sum of a finite geometric series proof
  48. Convergent sequence definition
  49. Partial sum definition
  50. Power series definition
  51. Binary relation definition
  52. Factorial formula for binomial coefficients statement
  53. Order on the reals definition
  54. Tautology definition
  55. Sum of a finite geometric series statement
  56. Radius of convergence definition
  57. Binomial series example
  58. Convergent series definition
  59. Antisymmetric relation definition
  60. Reflexive relation definition
  61. Transitive relation definition
  62. Proof of the binomial theorem by induction proof
  63. Logical equivalence definition
  64. Sum of a series definition
  65. Absolute convergence definition
  66. Binomial theorem statement
  67. Preorder definition
  68. Proof of the double negation law by a truth table proof
  69. Partial order definition
  70. Double negation law statement
  71. Total order definition
  72. Proof by contradiction concept
  73. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  74. The reals form an ordered field statement
  75. Absolute value definition
  76. Proof of the approximation property of the supremum proof
  77. Proof of the triangle inequality for real numbers proof
  78. Proof that the absolute value is multiplicative proof
  79. Proof of the epsilon-N definition of the limit of a sequence proof
  80. Approximation property of the supremum statement
  81. Triangle inequality statement
  82. Absolute value is multiplicative statement
  83. Epsilon-N definition of the limit of a sequence statement
  84. Proof that the reals are Archimedean from the completeness axiom proof
  85. Proof of the limit of a sum of sequences proof
  86. Proof of the squeeze theorem for sequences proof
  87. Proof that limits preserve non-strict inequalities proof
  88. Proof that convergent sequences are bounded proof
  89. The reals are Archimedean statement
  90. Proof of the monotone convergence theorem for sequences proof
  91. Proof that there is a natural number n with 1/n below any positive epsilon proof
  92. Limit of a sum of sequences statement
  93. Limits preserve non-strict inequalities statement
  94. Convergent sequences are bounded statement
  95. Proof of the ratio formula for the radius of convergence proof
  96. Squeeze theorem statement
  97. Monotone convergence theorem for sequences statement
  98. Proof of the limit of a product of sequences proof
  99. There is a natural number n with 1/n below any positive epsilon statement
  100. Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
  101. The sequence r^n tends to zero for |r| < 1 example
  102. Limit of a product of sequences statement
  103. Geometric series example
  104. A series of nonnegative terms converges iff its partial sums are bounded statement
  105. Proof of the ratio test proof
  106. Proof of the comparison test proof
  107. Comparison test statement
  108. Ratio test statement
  109. Ratio formula for the radius of convergence statement
  110. Proof that the binomial series has radius of convergence 1 proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Binomial_series, Conditions for convergence
About this page
Tag1JAFR3BDYN
URIm2://1JAFR3BDYN
Kindproof
WritingW0 stub
MSC30B10
ProvesRadius of convergence of the binomial series
UsesBinomial series, Ratio formula for the radius of convergence, There is a natural number n with 1/n below any positive epsilon, Epsilon-N definition of the limit of a sequence, Limit of a sum of sequences, Limit of a product of sequences
Changed4 November 2026

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