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| Knopp K. — Theory and applications of infinite series |
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| Предметный указатель |
"Almost all" 65
 200 230
, evaluation of 252—254
, series for 214 215
 -neighbourhood 20
 -function, Remann's 346 444—446 491—492 531 538
Abel — Dini theorem 290
Abel's convergence test 314
Abel's limit theorem 177 349
Abel's partial summation 313 397
Abel's series 122 281 292
Abel's theorem, extension of 406
Abel, N.H. 122 127 211 281 290 299 313 314 321 424 459 467 556
Abscissa of convergence 441
Absolute convergence of products 222
Absolute convergence of series 136 seqq. 396
Absolute value 7 390
Adams, J.C. 183 256
Addition 5 30 32
Addition theorem for the binomial coefficients 209
Addition theorem for the exponential function 191
Addition theorem for the trigonometrical functions 199 415
Addition, term by term 48 70 134
Aggregate, closed 7
Aggregate, ordered 5
Alterations, finite number of, for sequences 47 70 95
Alterations, finite number of, for series 130 476
Alternating series 131 250 263 316 518
Ames, L.D. 244
Amplitude 390
Analytic functions 401 seqq.
Analytic functions, series of 429
Andersen, A.F. 488
Approach within an angle 404
Approximation 65 231
Archimedes 7 104
Area 169
Arithmetic, fundamental laws of 5
Arithmetic, means 72 460
Arrangement by squares, by diagonals 90
Arzela, S. 344
Associative law 5 6
Associative law for series 132
Asymptotic series (expansion, representation) 518 seq. 535
Asymptotically equal 68
Asymptotically proportional 68 247
Averaged comparison 464—466
Axiom, Cantor — Dedekind 26 33
Axioms of arithmetic 5
Bachmann, F. 2
Barnes, E.W. 546
Bernoulli's inequality 18
Bernoulli's numbers 183 203—204 237 479
Bernoulli's polynomials 523 534
Bernoulli, James and John 18 65 184 238 244 457 523 556
Bernoulli, Nicolaus 324
Bertrand, J. 282
Bieberbach, L. 478
Binary fraction 39
Binomial series 127 190 208—211 423—428
Binomial theorem 50 190
Bocher, M. 350
Bohr, H. 492
Bolzano — Weierstrass theorem 91 394
Bolzano, B. 87 91 394
Bonnet, O. 282
Boormann, J.M. 195
Borel, E. 320 471 477 543 549 561 556
Bound 16 158
Bound, upper, lower 96 159
Bounded functions 158
Bounded sequences 16 44 80
Breaking off decimals 249
Briggs, H. 58 257
Bromwich, l'A. 477 556
Brouncker, W. 104
Burkhardt, H. 353 375 546
Cahen, E. 290 441
Cajori, F. 322
Cantor — Dedekind axiom 26 33
Cantor, G. 1 26 33 68 355
Cantor, M. 12
Carmichael, R.D. 477
Catalan, E. 247
Cauchy — Toeplitz limit theorem 74 391
Cauchy's convergence theorem 120
Cauchy's double series theorem 143
Cauchy's inequality 408
Cauchy's limit theorem 72
Cauchy's product 147 179 488 512
Cauchy, A.L. 19 72 87 96 104 113 117 136 138 146—148 164 186 196 219 294 408 459 534 545 556
Centre of a power series 157
Cesaro, E. 292 318 322 466
Chapman, S. 477
Characteristic of a logarithm 58
Circle of convergence 402
Circular functions 59 see
Closed aggregate 7
Closed expressions for sums of series 232—240
Closed interval 20 162
Commutative law 5 6
Commutative law for products 227
Commutative law for series 138
Comparison tests of the first and second kinds 113 seq. 274
Completeness of the system of real numbers 34
complex numbers see "Numbers"
Condensation test, Cauchy's 120 297
Conditionally convergent 139 226
Conditions F 464
Continued fractions 105
Continuity 161—162 171 174 404
Continuity of power series 174 177
Continuity of the straight line 26
Continuity, uniform 162
Convergence 64 78
Convergence for sequences 78—88
Convergence for series 110—120 124 282—290
Convergence for series of complex terms 396—401
Convergence for series of monotonely diminishing terms 120—124 294—296
Convergence for series of positive terms 116 117
Convergence for uniform convergences 332—338
Convergence of products 218 222
Convergence of series 101
Convergence, abscissa of 441
Convergence, absolute 136 seq. 222 396 435
Convergence, circle of 402
Convergence, conditional, unconditional 139 227
Convergence, criteria of see "Convergence tests" "Main
Convergence, general remarks on theory of 298—305
Convergence, half-plane of 441
Convergence, interval of 153 327
Convergence, radius of 151 seqq.
Convergence, rapidity of 251 262 279 332
Convergence, region of 153
Convergence, systematization of theory of 305—311
Convergence, tests for Fourier series 361 364—372
Convergence, uniform 326 seq. 381 428
Convergent sequences see "Sequences"
Cosine 199 seq. 384 414
Cotangent 202 seq. 417
Curves of approximation 329 330
D'Alembert, J. 458 459
De Morgan, A. 281
Decimal fractions 116 see
Decimal section 24 51
Dedekind's Test 315 348
Dedekind, R. 1 26 33 41
Dedekind, R., section 41
Dense 12
Diagonals, arrangement by 90
| Difference 31 243
Difference-sequence 87
Differentiability 163
Differentiability of a power series 174—175
Differentiability, right hand, left hand 163
Differentiation 163—164
Differentiation, logarithmic 382
Differentiation, term by term 175 342
Dini's rule 367—368 371
Dini, U. 227 282 290 293 311 344
Dirichlet series 317 441
Dirichlet's integral 356 seq. 359
Dirichlet's rule 365 371
Dirichlet's test 315 347
Dirichlet, G. Lejeune- 138 329 347 356 375 547
Disjunctive criterion 118 308 309
Distributive law 6 135 146
Divergence 65 101 160 391
Divergence, definite 66 101 160 391
Divergence, indefinite 67 101 160
Divergence, proper 67
Divergent sequences 457 seqq.
Divergent series 457 seqq.
Division 6 32
Division of power series 180 seqq.
Division, term by term 48 71
Divisors, number of 446 451 517
Divisors, sum of 451 547
Doetsch, G. 478
Double series, analogue for products 437—438
Double series, theorem on 430
du Bois-Reymond's test 315 348
du Bois-Reymond, P. 68 87 96 301 304 305 353 355 379 556
Duhamel, J.M.C. 285
e 82 194—198
e, calculation of 251
Eisenstein, G. 180
Elliot, E.B. 314
Equality 28
Equivalence theorem of Knopp and Schnee 481
Ermakoff's test 296 seqq. 311
Error 65
Error, evaluation of see "Evaluation of remainders"
Euclid 7 14 20 69
Eudoxus, postulate of 11 27 34
Eudoxus, theorem of 7
Euler's  -function 451 548
Euler's constant 225 228 271 522 527 536 538 547 555
Euler's formulae 353 415 518 536
Euler's numbers 239
Euler's transformation of series 244—246 262—265 469 507
Euler, L. 1 82 104 182 193 204 211 228 238 243 244 262 353 375 384 385 413 415 439 445 457 468 507 518 535—536 556
Evaluation of 252—254
Evaluation of e 251
Evaluation of logarithms 198 254—257
Evaluation of remainders 250 525 531—535
Evaluation of remainders, more accurate 259
Evaluation of roots 257—258
Evaluation of trigonometrical functions 258—259
Evaluation, numerical 247—260
Even functions 173
Everywhere convergent 153
Exhaustion, method of 69
Expansion of elementary functions in partial fractions 205—208 239 377 419
Expansion of infinite products 437
Expansion problem for asymptotic series 543 seqq.
Exponential function and series 148 191—198 411—414
Expressions for real numbers 230
Expressions for sums of series 230—273
Expressions for sums of series, closed 232—240
Extension 11 34
Faber, G. 546 556
Fabry, E. 267 556
Faculty series 446 seq.
Fatzius, N. 244
Fejer's integral 494
Fejer's theorem 493
Fejer, L. 493 496 546
Fibonacci's sequence 14 270 452
Finite number 15 16
Finite number of alterations see "Alterations"
Fourier, J.B. 352 375
Fourier, J.B., coefficients, constants 354 361 362
Fourier, J.B., Riemann's theorem on 363
Fourier, J.B., series 350 seqq. 492
Frobenius, G. 184 490
Frullani 375
Fully monotone 263 264 305
Function 158 403
Function, interval of definition, limit, oscillation, upper and lower bounds of 158—159
Functions of a complex variable 403 seq.
Functions of a real variable 158 seq.
Functions, analytic 401 seq.
Functions, arbitrary 351—352
Functions, cyclometrical 213—215 421
Functions, elementary 189 seq.
Functions, elementary analytic 410 seq.
Functions, even, odd 173
Functions, integral 408 411
Functions, rational 189 seq. 410
Functions, regular 408
Functions, sequences of 326 seq. 429
Functions, trigonometrical 198 seq. 258 414
Fundamental law of integers 7
Fundamental law of natural numbers 6—7
Fundamental law of order 5 29
Fundamental laws of arithmetic 5 32
Gamma-function 225—226 385 440 530
Gaps in the system of rational numbers 3 seqq.
Gauss, K.F. 1 113 177 288 289 552 556
Geometric series see "Series"
Gibbs' phenomenon 380 496
Glaisher, J.W.L. 180 555
Gmeiner, J.A. 399 556
Goldbach 458
Goniometry 415
Grandi, G. 133
Graphical representation 8 15 20 390
Gregory, J. 65 214
Gronwall, T.H. 380
Hadamard, J. 154 299 301 314
Hagen, J. 182
Hahn, H. 2 305
Half-plane of convergence 441
Hanstedt, B. 180
Hardy, G.H. 318 322 407 444 477 486 487 547 548
Harmonic series see "Series"
Hausdorff, F. 478
Hermann, J. 131
History of infinite series 104
Hobson, E.W. 350
Holder, O. 465 490
Holmboe 459
Horn, J. 550
Hypergeometric series 289
Identically equal 16
Identity theorem for power series 172
Improper integral see "Integral"
Induction, law of 6
Inequalities 7
Inequality of nests 29
Infinite number 15
Infinite series see "Series"
Infinitely small 19
Innermost point 23 394
Integrability in Riemann's sense 166
integral 165 seq.
Integral test 294
Integral, improper 169—170
Integral, logarithmic 545
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