xii
CONTENTS
CHAPTER XVII
SERIESSECTION PAGE 134. Introduction................................................................................................................................................................................................................................................................................................................................................................................................212 135. Infinite series................................................................................................................................................................................................................................................................................................................................................................................................213 136. Existence of a limit................................................................................................................................................................................................................................................................................................................................................................................................215 137. Fundamental test for convergence................................................................................................................................................................................................................................................................................................................................................................................................216 138. Comparison test for convergence................................................................................................................................................................................................................................................................................................................................................................................................217 139. Cauchy's ratio test for convergence................................................................................................................................................................................................................................................................................................................................................................................................218 140. Alternating series................................................................................................................................................................................................................................................................................................................................................................................................220 141. Absolute convergence................................................................................................................................................................................................................................................................................................................................................................................................220 142. Power series................................................................................................................................................................................................................................................................................................................................................................................................223 CHAPTER XVIII
EXPANSION OF FUNCTIONS143. Introduction................................................................................................................................................................................................................................................................................................................................................................................................227 144. Taylor's Theorem and Taylor's Series................................................................................................................................................................................................................................................................................................................................................................................................228 145. Maclaurin's Theorem and Maclaurin's Series................................................................................................................................................................................................................................................................................................................................................................................................230 146. Computation by series................................................................................................................................................................................................................................................................................................................................................................................................234 147. Approximate formulas derived from series. Interpolation................................................................................................................................................................................................................................................................................................................................................................................................237 148. Taylor's Theorem for functions of two or more variables................................................................................................................................................................................................................................................................................................................................................................................................240 149. Maxima and minima of functions of two independent variables................................................................................................................................................................................................................................................................................................................................................................................................243 CHAPTER XIX
ASYMPTOTES. SINGULAR POINTS150. Rectilinear asymptotes................................................................................................................................................................................................................................................................................................................................................................................................249 151. Asymptotes found by method of limiting intercepts................................................................................................................................................................................................................................................................................................................................................................................................249 152. Method of determining asymptotes to algebraic curves................................................................................................................................................................................................................................................................................................................................................................................................250 153. Asymptotes in polar coördinates................................................................................................................................................................................................................................................................................................................................................................................................254 154. Singular points................................................................................................................................................................................................................................................................................................................................................................................................255 155. Determination of the tangent to an algebraic curve at a given point by inspection................................................................................................................................................................................................................................................................................................................................................................................................255 156. Nodes................................................................................................................................................................................................................................................................................................................................................................................................258 157. Cusps................................................................................................................................................................................................................................................................................................................................................................................................259 158. Conjugate or isolated points................................................................................................................................................................................................................................................................................................................................................................................................260 159. Transcendental singularities................................................................................................................................................................................................................................................................................................................................................................................................260 CHAPTER XX
APPLICATIONS TO GEOMETRY OF SPACE160. Tangent line and normal plane to a skew curve whose equations are given in parametric form................................................................................................................................................................................................................................................................................................................................................................................................262 161. Tangent plane to a surface................................................................................................................................................................................................................................................................................................................................................................................................264
