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| Knopp K., Bagemihl F. — Infinite Sequences and Series |
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| Предметный указатель |
Real variable 16
Rearrangement of a sequence 22
Rearrangement of a series 53 79—84
Rearrangement theorem, extended 84
Reciprocal of a power series 115
Recursion formula 117
Regular function 102
Reitwiesner, G. W. 165
Remainder 2 29 50 164
Remainder, estimation of 164 165
Removal of parentheses 75
Reorder 28
Representation of a function 102
Roots, calculation of 167
Row series 84
Row-condition 35
Row-finite matrix 35
Rule, Cramer’s 117
Rules of operation, fundamental 75
Rules of parentheses 11
Rules, derived 11
Same order 30
Scales of convergence tests 128
Scales of divergence tests 129
Second main test for sequences 43
Second main test for series 67
Segment, partial 67
Semiopen interval 13
SEQUENCE 1 18
Sequence of bounded variation 72
Sequence of natural numbers 18
Sequence of numbers 25
Sequence, absolutely convergent 72
Sequence, arbitrary 18
Sequence, bounded 19
Sequence, boundedly divergent 30
Sequence, complex 18
Sequence, convergent 29
Sequence, decomposition of 22
Sequence, decreasing 11 19
Sequence, definitely divergent 30
Sequence, divergent 30
Sequence, double 28
Sequence, increasing 11 19
Sequence, indefinitely divergent 30
Sequence, limit of convergent 29
Sequence, limit point of 26
Sequence, monotonic 38
Sequence, monotonically decreasing 19
Sequence, monotonically increasing 19
Sequence, nondecreasing 19
Sequence, nonincreasing 19
Sequence, null 19
Sequence, oscillating 30
Sequence, positive 52
Sequence, real 18
Sequence, rearrangement of 22
Sequence, term of 1 26
Sequence, total variation of 72
Sequence, unboundedly divergent 30
Sequences, convergence tests for 38
Sequences, main tests for 37 38 43
Sequences, test for arbitrary 43
Sequences, test for monotonic 38
Series 2 44
Series of arbitrary terms, second main test for 67
Series of positive monotonically decreasing terms 61
Series of positive monotonically decreasing terms, condensation test for 62
Series of positive monotonically decreasing terms, integral test for 64
Series of positive terms 52 125
Series of positive terms, comparison tests for 52 55 56 57 132
Series of positive terms, first main test for 52
Series of positive terms, radical test for 57
Series of positive terms, ratio test for 57 58 129
Series transformation 88 140 170
Series transformation, Eider’s 144 171
Series transformation, Mark’s 88 172
Series transformation, Rummer’s 171
Series, absolutely convergent 71
Series, alternating 68
Series, binomial 140 158 159
Series, Cauchy product of 90 145
Series, closed evaluation of 163 168
Series, column 84
Series, conditionally convergent 82
Series, convergent 44
Series, convergent, addition of 78
Series, convergent, multiplication of 89 145
Series, convergent, operating with 74
Series, convergent, subtraction of 78
Series, definitely divergent 45
Series, dilution of 51 82
Series, Dirichlet 138
Series, divergent 45
Series, divergent, summation of 145
Series, double 86
Series, factor 90
Series, geometric 54 72
Series, harmonic 47
Series, harmonic, with exponent  54
Series, indefinitely divergent 45
Series, iterated 86
Series, logarithmic 158
Series, Maclaurin 108
Series, main tests for 52 67
Series, nonabsolutely convergent 71
Series, numerical evaluation of 163 165
Series, operating with 45 52
Series, power 98 99
Series, power, operating with 110
Series, rearrangement of 53 79-84
Series, row 84
Series, sum of 45
Series, Taylor 108
Series, term of 3
Series, unconditionally convergent 81
set of complex numbers 19
Set of numbers 12 25
Set of points 12
Set of real numbers 13 19
Set of real numbers, bounded 14
Set, boundary point of 13
Set, bounded 13
Set, closed 13
Set, element of 12 26
| Set, empty 12
Set, enumerable 27
Set, infinite 12
Set, interior point of 13
Set, isolated point of 13
Set, limit point of 13
Set, nonenumerable 27
Set, open 13
Set, point 19
Set, power of 27
Set, unbounded 13
Silverman, L.L. 35
Slower convergence 128
Slower divergence 128
Squares, arrangement by 28 90
Squares, principle of nested 39
Stage 20
Strength of divergence 66
Strictly decreasing 19
Strictly increasing 19
Stronger divergence 128
Subsequence 22
Subseries 53
Subset 12
Subset, proper 13
Subtraction of convergent series 78
Sum of a series 45
Sum, column 106
Sum, partial 45
Summation of divergent series 145
Summation, Abel’s partial 70 130
Summation, index of 4
Summation, interchange of order of 87
Sup 14
System of numbers 6
System of rational numbers, extension of 7
System of real numbers 7
System of real numbers, completeness theorem for 7
System of real numbers, uniqueness theorem for 7
Taylor expansion 108
Taylor series 108
Taylor series, Maclaurin form of 108
Taylor’s Theorem 17 149
Tend to a limit 29
Term of a product 94
Term of a sequence 1 26
Term of a series 3
Test, Abel’s 136 137
Test, condensation 62
Test, Dedekind’s 137
Test, Dirichlet’s 136 137
Test, Gauss’s 136
Test, integral 64
Test, Leibniz’s 68 138
Test, Raabe’s 136
Test, radical 57 59
Test, ratio 57 58 129
Tests for sequences, convergence 38 43
Tests for sequences, main 38 43
Tests for series, main 52 67
Tests, comparison 52 55 56 57 132 137
Theorem for power series, identity 103
Theorem of Cauchy 90
Theorem of continuity 7
Theorem of Dedekind 7
Theorem on finitely many alterations 50
Theorem on the transformation to a new center 105
Theorem, Abel’s limit 109 145 169
Theorem, addition 92 152 153
Theorem, best possible 142
Theorem, Bolzano — Weierstrass 15 42
Theorem, Cauchy — Toeplitz 73
Theorem, Cauchy’s double-series 172
Theorem, Cauchy’s—limit 33 37 110 144
Theorem, completeness 7
Theorem, extended rearrangement 84
Theorem, extension of binomial 109 151
Theorem, Taylor’s 17 149
Theory, function 17
Toeplitz — Cauchy theorem 73
Toeplitz, O. 35
Total variation 72
Transformation to a new center 105
Transformation, convergence-preserving 74
Transformation, Euler’s 144 171
Transformation, Kummer’s 171
Transformation, Markoff’s 88 172
Transformation, permanent 74 140
Transformation, series 88 140 170
Triangular matrix 35
Trigonometric functions 150 152
Trigonometric functions, calculation of 168
Trigonometric functions, inverse 160
Trigonometric representation of a complex numbBer 10
Unbounded set 13
Unboundedly divergent sequence 30
Unconditionally convergent series 81
Undetermined coefficients 117
Uniqueness theorem for the real number system 7
Unit circle 99
Upper bound 14
Upper bound, least 14 26
Upper limit 16 27
Value of a product 93
Value of a sequence, limiting 26
Value of a series 3 45
Value, absolute 10
Value, approximate 1
Value, principal, of arc sin z 160
Value, principal, of arc tan z 161
Value, principal, of the amplitude 10
Value, principal, of the general power 158
Value, principal, of the natural logarithm 156
Values, domain of 17 154 155
Vanishes 5
Variable, complex 16
Variable, real 16
Variation, bounded 72
Variation, total 72
von Mangoldt, H. 175
von Neumann, J. 165
Weaker divergence 128
Weierstrass — Bolzano theorem 15 42
Weimtrass, K. 132
Worse convergence 128
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