rational expression in x, 35
rational functions, 195
asymptotes of, 221 - 222
continuity of, 228
limits of, 195, 221-222
rational numbers, 4, 28
countability of, 127
denseness in R , 33, 76
of an ordered field F, 28
rational subfield, 28
real numbers, 42
uncountability of, 128
rearrangement
of a sequence, 72
of a series, 482
refinement of a partition, 361
regular partition of an interval, 379
relative complement of sets, 614, 615
restricted domain, 246
Riemann integral
Darboux's definition, 364
Riemann sum, 374
Riemann's criterion for
integrability, 368, 382
Riemann/Darboux criterion for
integrability, 372
Rolle's theorem, 321
root test, 468saltus of a function, 292, 444
second category set, 293
second derivative test, 328
sequence of partial sums of a series,
453
sequence( s) of real numbers, 50
bounded, 64
bounded away from zero, 65
Cauchy, 117
cluster point of, 111
constant, 62
contractive, 120
convergent, 52
divergent, 52
divergent to infinity, 82, 100
Index 735eventually constant, 63
eventually monotone, 104
Fibonacci, 121
limit of, 52, 61, 64
monotone increasing/ decreasing,
89
slowly divergent, 100
strictly increasing/ decreasing, 89
strictly monotone, 90
subsequence of, 107
summable, 501
absolutely s., 501
square s., 501
sequential criterion
for calculating integrals, 366, 377
for closed sets, 151
for cluster points, 150
for compactness, 160, 247
for continuity, 227
for denseness, 151
for differentiability, 300
for discontinuity, 227
for limits of functions, 184, 193
for one-sided continuity, 238
for one-sided limits, 206
for open sets, 153
for uniform continuity, 262
series
absolutely convergent, 480
alternating, 477, 478
alternating harmonic, 478, 486
binomial, 523
Cauchy product of, 488
conditionally convergent, 480
derived, 509
geometric, 454
harmonic, 456
hypergeometric, 4 76
Maclaurin, 512
nonnegative, 459
p-series, 456
power, 504
Taylor, 512
telescoping, 456