392 INDEX

convergence, 87recurrence relation, 58recursively defined sequence, 82recursively defined sequence, 58region, 270regular measures, 331regular Sturm Liouville problem, 225relatively closed, 105relatively open, 105relatively prime, 22removable discontinuity, 101removable singularity, 260residue, 264residues

method, 264Rieman Stieltjes integral

existence, 202Riemann integral

improper, 231without using sums, 155

Riemann Lebesgue lemma, 236Riemann Stieltjes integral

existence, 197properties, 196uniqueness, 196

Rolle’s theorem, 146root test, 94, 95roots

existence, 28roots of complex number, 184Russell’s paradox, 45

Schroder Bernstein theorem, 47second derivative test, 154second mean value theorem, 226

integrals, 239separated, 110sequence

functions, 116sequence of partial sums, 85sequences, 57

terminating, 53uncountable, 53

sequential compactness, 114sequentially compact

closed and bounded, 70in complex numbers, 71

series

absolute convergence, 87conditional convergence, 87convergence criterion, 87double sum, 96meaning of convergence, 85multiplication of series, 98nonnegative terms, 85p series test, 90ratio test, 94root test, 95

series of functionsuniform convergence, 119

sets, 43sigma algebra, 323simple function, 333Simpson’s rule, 223, 224smooth and not analytic, 179smooth curve, 251smoothness not analytic, 178space filling curve, 124squeezing theorem, 130squeezing theorem, 63star shaped, 256step function, 193Stirling’s formula, 215, 356

real arguments, 249sup

increasing sequence, 97interchange of order, 28

symmetric polynomial theorem, 364symmetric polynomials, 363

tagged division, 303Tarski, 53Taylor series, 157, 258

coefficients, 161convergence and divergence, 157differentiation, 159exponential sine and cosine, 162multiplication, 179of quotient, 188raising to a power, 186uniqueness, 161

Taylor’s formula, 148, 224Tietze extension theorem, 126total variation, 194totally ordered, 50trapezoid rule, 224