{VERSION 14 0 "Windows 7" "14.0" } {USTYLETAB {PSTYLE "Ordered List 1" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Ordered List 2" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }{PSTYLE "Ordered List 5" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 } 1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Annotation Title" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "C ourier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printe d Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "R3 Font \+ 2" -1 206 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 128 128 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE " " -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "D ash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkError" -1 207 1 {CSTYLE "" -1 -1 "Courier New" 1 12 255 0 255 1 2 2 1 2 2 1 1 0 0 1 } 1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "R3 Font 0" -1 208 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 128 0 128 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Ti mes" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 } {PSTYLE "HyperlinkWarning" -1 209 1 {CSTYLE "" -1 -1 "Courier New" 1 12 0 0 255 1 2 2 1 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Left Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{CSTYLE "Annotation Tex t" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Mapl e Name" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2 D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "H elp Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE " Help Normal" -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE " Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hel p Nonterminal" -1 24 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Variable" -1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Cour ier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Caption Reference" -1 201 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Pla ceholder" -1 202 "Courier New" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "Code" -1 203 "Courier New" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inert Output" -1 204 "Times" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 205 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Header and Footer" -1 206 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 207 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Label" -1 208 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Text" -1 209 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 212 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 213 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 210 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "" -1 211 "Ti mes" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 212 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "" -1 213 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 214 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "" -1 215 "Times" 1 14 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 216 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{PSTYLE "" -1 214 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 215 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 216 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 217 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 218 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "" -1 219 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 220 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 221 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE " " -1 222 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 223 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 224 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 225 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 217 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "" -1 218 "Times" 1 14 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{PSTYLE "" -1 226 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 227 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 228 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "" -1 229 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 230 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 231 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE " " -1 219 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "" -1 220 " Times" 1 14 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{PSTYLE "" -1 232 1 {CSTYLE " " -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 233 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 234 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 } 1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 235 1 {CSTYLE "" -1 -1 "Co urier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 236 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 237 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 238 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {CSTYLE "" -1 221 "Times" 1 14 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{PSTYLE "" -1 239 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 240 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 241 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 242 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 243 1 {CSTYLE "" -1 -1 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 210 "" 0 "" {TEXT 207 16 "Dr. Wei-Chi Yang" }} {PARA 211 "" 0 "" {TEXT 207 18 "Radford University" }}{PARA 212 "" 0 " " {TEXT 207 21 "www.radford.edu/wyang" }}{PARA 213 "" 0 "" {TEXT 207 22 "Pointwise Convergence." }}{PARA 0 "" 0 "" {TEXT 222 13 "A sequence \{ " }{TEXT 222 45 "\} of functions from a set S to R is said to " } {TEXT 210 18 "converge pointwise" }{TEXT 222 52 " to a function f if for every x in S we have \{" }{TEXT 222 48 "\} converges to f(x) as n tends to infinity. \n" }}{PARA 0 "" 0 "" {TEXT 211 19 " Bounded Convergence" }{TEXT 222 1 "\n" }{TEXT 222 2 " \n" } {TEXT 222 12 "A sequence \{" }{TEXT 222 49 "\} of functions from a se t S to R is said to " }{TEXT 212 18 "converge boundedly" }{TEXT 222 22 " to a function f if \{" }{TEXT 222 82 "\} converges pointwis e to f and it is possible to find a number K such that |" }{TEXT 222 39 "| <= K for every x in S and every n. \n" }}{PARA 0 "" 0 "" {TEXT 213 19 "Uniform Convergence" }{TEXT 222 1 "\n" }{TEXT 222 2 " \n" }{TEXT 222 12 "A sequence \{" }{TEXT 222 48 "\} of func tions from a set S to R is said to " }{TEXT 214 18 "converge unifo rmly" }{TEXT 222 28 " to a function f if sup |" }{TEXT 222 3 " - " }{TEXT 215 4 "f(x)" }{TEXT 222 42 " | tends to 0 as n tends to infi nity. \n" }}{PARA 0 "" 0 "" {TEXT 216 10 "Exercise 1" }{TEXT 222 46 ": Demonstrate that the sequence of functions " }{TEXT 222 58 "is pointwise convergent to 0 but not uniformly convergent." }}{PARA 214 "> " 0 "" {MPLTEXT 1 0 25 "f:=(x,n)-> n*x*exp(-n*x);" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 7 "f(x,n);" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 0 41 "plot(\{f(x,10^6),f(x,10^7)\},x=0..10^(-5));" }}} {EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 218 "> " 0 "" {MPLTEXT 1 0 42 "plot(\{f(x,10^9),f(x,10^10)\},x=0..10^(-8));" }} }{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 0 35 "P:=(x,k)-> (10^k)*x*exp(( -10^k)*x);" }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 0 47 "plot(\{P(x, 12),P(x,11)\},x=0..10^(-10),y=0..0.4);" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 222 "> " 0 "" {MPLTEXT 1 0 27 "for k from 9 by 1 to 16 do " }}{PARA 223 "> " 0 "" {MPLTEXT 1 0 36 "P||k:= plot (\{P(x,16),P(x,k)\}, x=0.." }{MPLTEXT 1 0 8 "10^(-10 )" }{MPLTEXT 1 0 25 ", y=0..0.40,thickness=2):" }}{PARA 224 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 0 54 "pl ots[display]([seq(P||k, k=9..16)], insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 10 "Exercise 2" }{TEXT 218 1 ":" }{TEXT 222 58 " D emonstrate by animation that the sequence of functions " }{TEXT 222 25 "converges uniformly to 0." }}{PARA 0 "" 0 "" {TEXT 222 0 "" }}} {EXCHG {PARA 226 "> " 0 "" {MPLTEXT 1 0 35 "f:=proc(x,k) k*x*exp((-k^2 )*x) end;" }}}{EXCHG {PARA 227 "> " 0 "" {MPLTEXT 1 0 12 "with(plots): " }}}{EXCHG {PARA 228 "> " 0 "" {MPLTEXT 1 0 27 "for k from 1 by 1 to \+ 20 do " }}{PARA 229 "> " 0 "" {MPLTEXT 1 0 61 "f||k:= plot (\{f(x,20), f(x,k)\}, x=0..1, y=0..0.6,thickness=2):" }}{PARA 230 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 231 "> " 0 "" {MPLTEXT 1 0 54 "pl ots[display]([seq(f||k, k=1..20)], insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 219 10 "Exercise 3" }{TEXT 220 1 ":" }{TEXT 222 56 " D emonstrate by animation that the sequence of functions" }{TEXT 222 39 " converges pointwise but not boundedly." }}{PARA 0 "" 0 "" {TEXT 222 0 "" }}}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 0 33 "f:=proc(x,k) k^2*x *exp(-k*x) end;" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 0 12 "with(p lots):" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 0 27 "for k from 1 by 1 to 30 do " }}{PARA 235 "> " 0 "" {MPLTEXT 1 0 60 "f||k:= plot (\{f( x,30),f(x,k)\}, x=0..1, y=0..15,thickness=2):" }}{PARA 236 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 0 54 "pl ots[display]([seq(f||k, k=1..30)], insequence=true);" }}}{EXCHG {PARA 238 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 221 11 "Exercise 4:" }{TEXT 222 56 " Uniform convergence on functions and \+ differentiability:" }}{PARA 0 "" 0 "" {TEXT 222 148 "We have a sequenc e of functions f(x,k) defined on [-1,1] below, we note that f(x,k) con verges uniformly to 0 in [-1,1] but \{f'(0,k)\} converges to 1." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 207 0 "" }}}{EXCHG {PARA 239 "> " 0 "" {MPLTEXT 1 0 38 "f:=(x,k)-> x/(1+k*x^2); " }}}{EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 0 30 "for k from 150 by 1 to 200 do " } }{PARA 241 "> " 0 "" {MPLTEXT 1 0 53 "f||k:= plot (\{f(x,200),f(x,k)\} , x=-1..1,thickness=2):" }}{PARA 242 "> " 0 "" {MPLTEXT 1 0 3 "od:" }} }{EXCHG {PARA 243 "> " 0 "" {MPLTEXT 1 0 57 "plots[display]([seq(f||k, k=150..200)], insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }