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Concept Based Notes Maths II B.Sc I Year Biyani's Think Tank Concept based notes Maths II B.Sc I Year Megha Sharma Department of Science Biyani’s Group of Colleges 2 Published by : Think Tanks Biyani Group of Colleges Concept & Copyright : Biyani Shikshan Samiti Sector-3, Vidhyadhar Nagar, Jaipur-302 023 (Rajasthan) Ph : 0141-2338371, 2338591-95 Fax : 0141-2338007 E-mail : [email protected] Website :www.gurukpo.com; www.biyanicolleges.org ISBN: Edition : 2013 While every effort is taken to avoid errors or omissions in this Publication, any mistake or omission that may have crept in is not intentional. It may be taken note of that neither the publisher nor the author will be responsible for any damage or loss of any kind arising to anyone in any manner on account of such errors and omissions. Leaser Type Setted by : Biyani College Printing Department For free study notes log on: www.gurukpo.com Maths 3 Preface am glad to present this book, especially designed to serve the needs of the I students. The book has been written keeping in mind the general weakness in understanding the fundamental concepts of the topics. The book is self-explanatory and adopts the “Teach Yourself” style. It is based on question-answer pattern. The language of book is quite easy and understandable based on scientific approach. Any further improvement in the contents of the book by making corrections, omission and inclusion is keen to be achieved based on suggestions from the readers for which the author shall be obliged. I acknowledge special thanks to Mr. Rajeev Biyani, Chairman & Dr. Sanjay Biyani, Director (Acad.) Biyani Group of Colleges, who are the backbones and main concept provider and also have been constant source of motivation throughout this endeavour. They played an active role in coordinating the various stages of this endeavour and spearheaded the publishing work. I look forward to receiving valuable suggestions from professors of various educational institutions, other faculty members and students for improvement of the quality of the book. The reader may feel free to send in their comments and suggestions to the under mentioned address. Megha Sharma For free study notes log on: www.gurukpo.com 4 Syllabus For free study notes log on: www.gurukpo.com Maths 5 Unit – I Series Series : Infinite Series and convergent series: Series : Expression of that form in which the successive terms are always according to some definite rule. Here – nth term of series There are two types of series 1) Finite Series 2) Infinite Series Infinite Series: If the no. of term in any series is infinite then the series is called infinite series. Eg. it’s denoted by Convergent Series : An infinite series is said to be convergent if the sequence of its partial sum <Sn> is convergent = S (finite) n – d Test for convergence of series : 1) D’ Alembert’s Ratio test : If be a series of positive terms such that then (i) if l > 1, will be convergent (ii) ifl < l, will be divergent (iii) if l = 1 may either convergent or divergent Cauchy nth root test : If be a series of positive terms such that = (a real number) then (i) if <1, will be convergent (ii) if, >1 will be divergent (iii) if =1, may either converge or diverge For free study notes log on: www.gurukpo.com 6 Raabe’s tests : If be a series of positive terms such that then (i) if > 1, will be convergent (ii) if <1, will be divergent (iii) if =1, may either converge or diverge D’e Morgan and Bertrand’s Test : If be a series of positive terms such that then (i) if >1, will be convergent (ii) if <1, will be divergent Cauchy’s condensation test : If the series is a positive terms such that <f(n) is a decreasing sequence and if a >1 and is a positive integer, then the series and both converge or diverge together. Gauss tests : If be a series of positive terms and Where sequence <Yn> is bounded then (a) if (i) is convergent if (ii) is divergent if (b) if (i) is convergent if (ii) is divergent if < 1 Alternating Series : Any series of the type +… ( ) Where terms are alternatively positive and negative, is known as an alternating series and is denoted by Eg. 1 - - + …. For free study notes log on: www.gurukpo.com Maths 7 A series whose terms are alternatively positive or negative is aid to be absolutely convergent if the series both are convergent. Taylor’s theorem: (i) Taylor’s theorem with L1agrange’s form of Remainder Statement : if in the interval [a, a+h], a function f is defined in such a way that the differentials f’ , f” ,….., upto the order (n-1) are (i) Continuous in the interval [a, a+h], (ii) nth derivative of f exist in the interval (a, a+h) then there exist at least one number Q between O and 1, such that f(a+h) = f(a) + h f’ (a) + f” (a) + … + (iii) Taylor’s theorem with cauchy’s from of Remainder: Statement : if in the interval [a, a+h] a function f(x) is defined in such a way that (i) all derivatives of f (x) up to the order (n-1) are continuous in the interval [a, a+h] (ii) all derivatives of f (x) up to the order n exist in the interval (a, a+h), then there exist atleast one number between o and 1, such that. f(a+h) = f(a) + h f’(a) + f” (a) +…. Maclaurin’s theorem : Statement : if a function f in is Such that (i) f, f` …. is continuous in [ (ii) in (iii) (0,1) such that f = f(o) + x f’(o) + + ….+ (o)+ where Rn = is called maclaurin’s remainder due to schlomitch and Roche. Power Series If is a continuous real variable and ‘a’ any constant then the series is Constant, is called a power series about the point x =a For free study notes log on: www.gurukpo.com 8 is called Standard power series. Q.1 Show that the following series is divergent: Sol. Let = and = = = D ‘Alembert’s ratio test fail Again .n = = again Raabe’s test also fail again = = - = - = Therefore by De’ morgan and Bertrand’s test the series is divergent. Q.2 Discuss the convergence and absolute convergence of the following series Sol. Given series is alternating series Clearly and Un = For free study notes log on: www.gurukpo.com Maths 9 therefore by Leibnitz test the above series is also convergent. again = Let us take auxiliary series Vn = Now (non zero finite quantity) Both will converge or diverge together. But = is divergent, Because p = 1 is conditionally convergent. Q.3 Show that the following series is convergent if x < 1 and divergent if x> 1 + Sol. Un = Un+1 = Now. = . Therefore if convergent. and if Therefore at = = = at For free study notes log on: www.gurukpo.com 10 Q.4 Test for the convergence of the following series. Sol. Un = = = = (1+o) . 1.x = Therefore by Cauchy nth foot test the given series is divergent if > 1 convergent if < 1 if x =1 then Un = = let us take auxiliary series is where Vn = Now = e# 0 Therefore by comparison test will converge or diverse together. But therefore together is also divergent. Q.5 Test for the convergence and absolute convergence of the following series: +…… Case - I = when P > O Sol. Given series is alternating series and <Un-1 And again ∑ is convergent series = 1 If p> 1, series will be convergent If o<p< 1 series will be divergent. Therefore for p>1 the given series will be absolute convergent and o<p< 1, the given series will be conditionally convergent Case - II = When p = o In this case the given series ∑un = 1 – 1 + 1 – 1 + …. is For free study notes log on: www.gurukpo.com Maths 11 ______ oscillates finitely Case - III = When p < o Let p = m, where m > o then ∑un = ∑ = ∑ = 1 - + here Un = or when n simultaneous odd and even. and therefore given series is oscillating series between - Q.6 Test for the convergence of the following series. Sol. For the given series if we neglect first term of given series then convergence of given series not affected therefore we Let Then Un = and Un +1 = Now = = therefore if ∑un is convergent and if x > 1 then < 1 is divergent. If x = 1 then D’Almbert’s ratio test fail. at x = 1 n = ( ) = test given series is convergent. Therefore if < ! the given series will be convergent and the given series will be divergent. Q.7 Find Lagrange’s and Cauchy’s remainder after n terms in the expansion of following functions. For free study notes log on: www.gurukpo.com 12 (i) log (1+x) Sol. Let f ; possesses derivatives of every = ! = Q.8 If f(x) is continuous in [a,b] and possesses finite derivatives for x = C (a,b), then prove that Sol. Since the function is differentiable at x=c, hence f’ (x) and f’ (x) exist in the neighborhood (c-h, c+h) of x =c Now applying second mean value theorem for the intervals (c-h, c) and (c,c+h), we have f f(c-h)=f(c) – hf’(c) + O< <1 F(c+h) = f(c) + h’(c) + f” (c+ ) --------------(2) O< <1 Adding (1) and (2), we obtain F”(c- h) +f(c+h) = 2f(c) + - f(c-h)+ f(c+h) – 2f(c)= [f”(c- [ f”(c- ] Q.9 Test the convergence of following series: +….. Sol. Let first term of this series 0 = then f(n) = = For free study notes log on: www.gurukpo.com Maths 13 Suppose f = then Vn = nd Vn+1 = now = a>1 is convergent therefore by cauchy’s consensation test the given series is convergent.
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