Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

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302 Series [Ch. 8 sin na 2481. (_l)"l^, 2482. ( ly-'tan ^ n^i n V n yourself that 2483. Convince yourself that the d'Alembert test for convergence ide the quesl does not decide the question of the convergence of the series 2 a > where whereas by means of the Cauchy test it is possible to establish that this series converges. 2484*. Convince yourself that the Leibniz test cannot be applied to the alternating series a) to d). Find out which of these series diverge, which converge conditionally and which converge absolutely: _J___l a) ' | j " .j; 1 _1_ . . 1 3 +y 3F + 2 1 3?+ v i_ . ' 3 _ L __u *~ 2 3 3 " h 5 3* r " ' ' ' ! ] ! J ! ^\ i . i . d) T _l + _ T T + _ TT T +... Test the following series with complex terms for convergence: CO 00 2485. ^2+#. 2488. V. 00 00 ' (2 2486, X" 37 1) ''. 2489. 1=1 =i
Sec. 1]_Number Series_303 - V r(2-Q + l 1" [ fa [n(3-20-3ij ' 2493. Between the curves {/ = -? and y=-^ and to the right X X of their point of intersection are constructed to segments parallel the t/-axis at an equal distance from each other. Will the sum of the lengths of these segments be finite? 2494. Will the sum of the lengths of the segments mentioned in Problem 2493 be finite if the curve \)^-\ is replaced by the x curve y = ? X 2495. Form the sum of the series ^-^ and Does this sum converge? and QC 2496. Form the difference of the divergent series ]T 2 n i rt anc* * es * '* for convergence. 2497. Does the series formed by subtracting the series 00 sef i TT -f es 21 converge? n = i n i 2498. Choose two series such that their sum converges while their difference diverges. 00 00 CC QC 2499. Form the product of the series V ~nVn and V.OTTMntt 2 Does this product converge? 2500. Form the series +1 + 1+ .. . + . J~-f- . . V. (l Does this series converge? 2501. Given the series 1+1 1+ ...+ ( -: ^+... Estimate the error committed when replacing the sum of this series with the sum of the first four terms, the sum of the first five terms. What can you say about the signs of these errors? 2502*. Estimate the error due to replacing the sum of the series y + 2! (2") + ~3\('2J + " + 5H2") +" by the sum of its first n terms. r
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TEXT FLY WITHIN THE BOOK ONLY
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OUP-43 30-1.71 5,000 OSMANIA UNIVER
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T. C. BapaneHKoe. C. M. Koean, r Jl
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TO THE READER MIR Publishers would
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Contents Sec. 6. Integrating Certai
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8 Chapter Contents Sec. 13. Linear
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Sec. 1. Functions Chapter I INTRODU
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Sec 1] Functions 13 8. Find the rat
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Sec. 1]_Functions_15 34. Prove that
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Sec. 2] Graphs of Elementary functi
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Sec. 2] Graphs of Elementary functi
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Sec. 2] Graphs of Elementary Functi
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Sec. 3] Limits 23 For the existence
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Sec. 3] 179. Hm (Vn + 1 \f~n). n -+
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Sec. #] Limits 27 209. lim 210. lim
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^Sec 3] Limits Therefore, Example 9
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Sec. 31 Limits 31 269. a) 270. lim-
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Sec. 4] Infinitely Small and Large
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Sec. 4] Infinitely Small and Large
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Sec. 5] Continuity of Functions 37
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Sec. 5]_Continuity of Functions_39
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Sec. 5] Continuity of Functions 334
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Sec. 1] Calculating Derivatives Dir
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Sec. 1] Calculating Derivatives Dir
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Sec. 2]__Tabular Differentiation_47
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Sec. 2]_Tabular Differentiation_49
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Sec. 2] Tabular Differentiation_ 51
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Sec. 2] Tabular Differentiation 504
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Sec. 2] Tabular Differentiation 55
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Sec. 3] The Derivatives of Function
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Sec. 3] The Derivatives of Function
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Sec 4] Geometrical and Mechanical A
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Sec. 4\ Geometrical and Mechanical
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Sec. 4] Geometrical and Mechanical
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Sec. 5]_Derivatives of Higher Order
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Sec. 5]_Derivatives of Higher Order
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Sec. 6] Differentials of First and
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Sec. 6}_Differentials of First and
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Sec. 7]_Mean-Value Theorems_75 745.
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Sec. 8] Taylor's Formula 77 765. a)
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Sec. 9]_ V Hospital-Bernoulli Rule
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Sec. 9] L' Hospital-Bernoulli Rule
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Chapter III THE EXTREMA OF A FUNCTI
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Sec. 1] The Extrema of a Function o
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Sec. 2]_The Direction of Concavity.
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Sec. 3]_Asymptotes_93 Example 2. Fi
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Sec. 3]_Asymptotes_95 Solution. Sin
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Sec 4] Graphing Functions by Charac
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CM CM -h CO O non ' > C " -~ a - >
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Sec. />] Differential of an Arc. Cu
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Sec 5] Differential of an Arc. Curv
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Sec. 5]_Differential of an Arc. Cur
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Sec. 1. Direct Integration 1. Basic
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Sec 1] Direct Integration_109 1048*
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Sec. 2] Integration by Substitution
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Sec. 2] Integration by Substitution
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Sec. 3\_Integration by Parts_H7 Exa
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Sec 4] Standard Integrals Containin
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Sec. 5] Integration of Rational Fun
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Sec. 5] In t egration of Rational F
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Sec. 6] Integrating Certain Irratio
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Sec. 6] Integrating Certain Irratio
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Sec. 7]_Integrating Trigonometric F
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Sec. 7]_Integrating Trigonometric F
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Sec. 8] Integration of Hyperbolic F
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Sec. 11] Using Reduction Formulas 1
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Sec. 12] Miscellaneous Examples on
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Sec. 1] The Definite Integral as th
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Sec. 2]_Evaluating Definite Integra
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Sec. 3]_Improper Integrals_143 Sec.
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Sec. 3] Improper Integrals 145 Solu
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Sec. 4] Change of Variable in a Def
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Sec. 5] Integration by Parts 149 Bu
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Sec. 6] Mean-Value Theorem 151 In p
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Sec. 7] The Areas of Plane Figures
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Sec. 7] The Areas of Plane Figures
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Sec. 7\_The Areas of Plane Figures_
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Sec. 8] The Arc Length of a Curve 1
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Sec, 9] Volumes of Solids 161 1673.
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Sec. 9] Volumes of Solids 163' Solu
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Sec. 9]_Volumes of Solids_165 1693.
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Sec. 10] The Area of a Surface of R
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Sec. II] Moments. Centres of Gravit
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Sec 11] Moments. Centres of Gravity
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&?c. 12] Applying Definite Integral
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Sec. 12] Applying Definite Integral
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Sec. 1) Basic Notions [SI Putting *
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Sec. 1]_Basic Notions_183 1790*. Le
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Sec. 3] Partial Derivatives 185 180
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Sec. 4]_Total Differential of a Fun
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Sec. 4}_Total Differential of a Fun
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Sec. 5]_Differentiation of Composit
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Sec. 6]_Derivative in a Given Direc
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Sec. 6] Derivative in a Given Direc
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Sec. 7]_Higher-Order Derivatives an
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Sec. 7] Higher-Order Derivatives an
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Sec. 7]_Higher-Order Derivatives an
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Sec. ti\_Integration of Total Diffe
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Sec. 9]_Differentiation of Implicit
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Sec. 9] Differentiation of Implicit
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Sec. 9]_Differentiation of Implicit
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Sec. 10]_Change of Variables_2H 196
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Sec. 10]_Change of Variables_213 or
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Sec. 10] Change of Variables 215 an
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Sec. 11]_The Tangent Plane and the
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Sec. 11]_The Tangent Plane and the
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Sec. 12] Taylor's Formula for a Fun
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Sec. 13]_The Extremum of a Function
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Sec. 13] The Extremutn of a Functio
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Sec. 14] Finding the Greatest and S
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Sec. 14] Finding the Greatest and S
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Sec. 15] Singular Points of Plane C
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Sec. 16] Envelope 233 Solution. The
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Sec. 18] The Vector Function of a S
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Sec. 18]_The Vector Function of a S
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Sec. 19] The Natural Trihedron of a
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Sec. 19] The Natural Trihedron of a
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Sec. 20] Curvature and Torsion of a
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Sec. 20]_Curvature and Torsion of a
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Sec. 1] The Double Integral in Rect
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Sec. 1] The Double Integral in Rect
Page 257 and 258: Sec. 1] The Double Integral in Rect
Page 259 and 260: Sec. 2]_Change of Variables in a Do
Page 261 and 262: Sec. 2]_Change of Variables in a Do
Page 263 and 264: Sec 3]_Computing Areas_257 2177. Co
Page 265 and 266: Sec. 5]_Computing th Area* of Surfa
Page 267 and 268: Sec. 6]_Applications of the Double
Page 269 and 270: Sec. 7] Triple Integrals 2o3 Evalua
Page 271 and 272: Sec 7\ Triple Integrals 265 We ther
Page 273 and 274: gee. 7]_Triple Integrals_267 2251.
Page 275 and 276: Sec. 8] Improper Integrals Dependen
Page 277 and 278: Sec. 8]_Improper Integrals Dependen
Page 279 and 280: See. 9] Line Integrals 273 Test for
Page 281 and 282: Sec. 9]____Line Inlegtah____275 whe
Page 283 and 284: Sec. 9]_Line Integrals__277 express
Page 285 and 286: Sec. 9] Lim Integrals 279 (/-axis;
Page 287 and 288: Sec 9] Line Integrals 281 Evaluate
Page 289 and 290: Sec. 9] Line Integrals 2S& D. Appli
Page 291 and 292: Sec. 10] Surface Integrals 285 When
Page 293 and 294: ec. 11]_The Ostrogradsky-Gauss Form
Page 295 and 296: Sec. 12] Fundamentals of Field Theo
Page 297 and 298: Sec. 12] Fundamentals of Field Theo
Page 299 and 300: Sec. 1. Number Series Chapter VIII
Page 301 and 302: Sec. 1]_Number Series_295 c) D'Alem
Page 303 and 304: Sec /\_Number Series_297 Note. For
Page 305 and 306: 2418. + + 0419 J___1_ . 4- 4- 4- '
Page 307: Sec. 1] Number Series 2464. cos-2-)
Page 311 and 312: Sec. 2]_Functional Series_305 In th
Page 313 and 314: Sec. 2]_Functional Series_307 [the
Page 315 and 316: Sec. 2] Functional Series 309 Deter
Page 317 and 318: Sec. 3]_Taylor's Series_311 v8 s r
Page 319 and 320: Sec. 3] Taylor's Series 313 Since a
Page 321 and 322: Sec 3} Taylor's Series 315 2615. ln
Page 323 and 324: Sec. 3] Taylor's Series 317 2651. F
Page 325 and 326: Sec. 4] Fourier Series 319 If a fun
Page 327 and 328: Sec. 4\__Fourier Series _321 2690.
Page 329 and 330: Sec 1 1 Verifying Solutions 323 It
Page 331 and 332: Sec. 21 First-Order Differential Eq
Page 333 and 334: Sec. 3] Differential Equations with
Page 335 and 336: Sec. 3\ Differential Equations with
Page 337 and 338: Sec. 4]_First-Order Homogeneous Dif
Page 339 and 340: Sec. 5] Bernoulli's Equation 333 Co
Page 341 and 342: Sec. 6] Exact Differential Equation
Page 343 and 344: Sec. 7] First-Order Differential Eq
Page 345 and 346: Sec 8]_The Lagrange and Clairaut Eq
Page 347 and 348: Sec. 9] Miscellaneous Exercises on
Page 349 and 350: Sec. 9] Miscellaneous Exercises on
Page 351 and 352: Sec. 10] Higher-Order Differential
Page 353 and 354: Sec. 10]__Higher-Order Differential
Page 355 and 356: Sec. 11]_Linear Differential Equati
Page 357 and 358: Sec. 12] Linear Differential Equati
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Sec. 12] Linear Differential Equati
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Sec. 12] Linear Differential Equati
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Sec. 14]_Euler's Equations__357 Fin
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Sec. 15]_Systems of Differential Eq
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.Sec. 16} Integration of Differenti
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Sec. 17] Problems on Fourier's Meth
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Sec. 17]_Problems on Fourier's Meth
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Chapter X APPROXIMATE CALCULATIONS
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Sec. 1] Operations on Approximate N
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Sec. /]_Operations on Approximate N
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Sec, 2] Interpolation of Functions
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Sec. 2] Interpolation of Functions
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Sec. 3]_Computing the Real Roots of
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Sec. 4] Numerical Integration of Fu
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Sec. 5] Numerical Integration of Or
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Table 3. Basic Table for Calculatin
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394 Approximate Calculations \Ch 10
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ANSWERS Chapter I 1. Solution. Sinc
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398 Answers Form a table of values:
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400 Answer* 271. Solution. If x & k
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402 Answers b) /'(!)= lim 1^15^= li
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404_Answers 494. , . 495. - - . 1 2
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-406 Answers 640. Hint. The equatio
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408_Answers_ 755. e xcosa sin(*sina
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410_Answers_4 where R is the radius
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412 Answers 1/~JT f Q \ J/max = TT=
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414 Answers 976. j/mi n =1.32 when
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416 Answers 1037. j/ 1040. 3x* 13 J
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418 1 1137. 4- 6 3a . In I ocot3x|
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420 Answers 4^8 Problem 1218*. 1221
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422 Answers X arctan(* + 2). 1294.
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424 Answers sin 5* , 1374. -!- 1377
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426 Answers 3 1442. In l-x2 ' '" 2
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428_Answers_ the function /(x) = --
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430 Answers 1651. 3na 2 . 1652. n(b
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432 Answers inertia /=Y 2jttf 1 r *
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434 Answers along the large lower s
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436 Answers cos^. ,810. * ox x* x d
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438_Answers 1905. 1914. u(*. !/) =
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440 Answers 1983. 3* + 4j/-H2z 169
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442 Answers is ~ |/2 + -~=, the alt
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444 Answers (binormal). 2103. bx *=
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444_Answers_x = \ ' > (principal no
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446 Answers 2 a a a 2139. ( dy(f(x,
Page 456 and 457:
448_Answers_ r 4 **/* [ rj-cos*(p--
Page 458 and 459:
450 Answers 2243. 2244. 1 i VT=T* ^
Page 460 and 461:
452 Answers 2 t 2297. ) ~ll . |-[(I
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454_Answers__ TtD the flux, use the
Page 464 and 465:
456__Answers_ and when x | ^ =- , t
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458 Answers
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460 Answers %-l n=o (n)=^=%. 2873.
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462 Answers Chapter IX 2704. Yes. 2
Page 472 and 473:
464 Answers 2822. g-C*+; y=2*. 2824
Page 474 and 475:
466 Answers x + C z . 2914. #=
Page 476 and 477:
468_Answers_ A $007. 1) x = C cosco
Page 478 and 479:
470 Answers 3081. x = C l e 3082. x
Page 480 and 481:
472 Answers ,*. f (2"+ 1)JtJf dx. H
Page 482 and 483:
474 Answers 3131. a) 0.211; 0.389;
Page 484 and 485:
476 Appendix III. Inverse Quantitie
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478 Appendix IV. Trigonometric Func
Page 488 and 489:
480 Appendix VI. Some Curves (for R
Page 490 and 491:
482 Appendix 11. Graphs //-=secx an
Page 492 and 493:
486 Appendix 24. Strophoid, * u s^-
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Absolute error Absolute value 367 o
Page 496 and 497:
490 Index Differential equations of
Page 498 and 499:
492 Index Infinite discontinuities
Page 500 and 501:
494 Index Operator Hamiltonian 288
Page 502:
496 Index Taylor's series Term 311,
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