BT ( Binomial Theorem )

ATUL KUMAR ID-atul.mod@gmail .com mob:09310556579 ATUL KUMAR MATHEMATICS KEY CONCEPTS Binomial, Exponential and Logarithmic Series 1. Binomial Theorem: The formula by which any positive integral power of a binomial expression can be expanded in the form of a series is known as Binomial Theorem. If x, y ∈ R and n ∈ N, then; (x + y)n = nC0 xn + nC1 xn -1 y + nC2 xn – 2 y2 + …….. + nCr xn – r yr + …….. + nCn yn∑=−n0rrrnrnyxC = . This theorem can be proved by induction. Observations: (i) The number of terms in the expansion is (n + 1) i.e., one or more than the index. (ii) The sum of the indices of x and y in each term is n. (iii) The binomial coefficients of the terms nC0, nC12. Important Terms In The Binomial Expansion Are: ……… equidistant from the beginning and the end are equal. (i) General term (ii) Middle term (iii) Term independent of x & (iv) Numerically greatest term (i) The general term or the (r + 1)th term in the expansion of (x + y)n is given by; Tr + 1 = nCr xn – r . yr(ii) The middle term(s) is the expansion of (x + y). n(a) If n is even, there is only one middle term which is given by; is (are): T(n + 1)/2 = nCn/2 . xn/2 . y(b) If n is odd, there are two middle terms which are: n/2 T(n + 1)/2 & T(iii) Term independent of x contains no x; Hence find the value of r for which the exponent of x is zero. [(n + 1)/2] + 1 (iv) To find the Numerically greatest term is the expansion of (1 + x)nxr1rnxCxCTT1r1rnrrnr1r+−==−−+, n ∈ N find . Put the absolute value of x and find the value of r. Consistent with inequality 1TTr1r>+. Note that the Numerically greatest term in the expansion of (1 – x)n, x > 0, n ∈ N is the same as the greatest term in (1 + x)n3. If . ()fIBAn+=+, where I and n are positive integers, n being odd and 0 < f < 1, then (I + f). f = Kn where A – B21BA<− = K > 0 and . If n is an even integer, then (I + f) (1 – f) = Kn4. Binomial Coefficients: . (i) C0 + C1 + C2 + ……… + Cn = 2(ii) Cn 0 + C2 + C4 + ………. = C1 + C3 + C5 + ……… = 2(iii) Cn – 1 02 + C12 + C22 ……… + Cn2 = 2nCn()!n!n!n2 = (iv) C0 . Cr + C1 . Cr + 1 + C2 . Cr + 2 + ………. + Cn – r . Cn()()()!rnrn!n2−+ = Remember: (i) (2n)! = 2n5. Binomial Theorem For Negative Or Fractional Indices: . n! [1. 3. 5 …….. (2n – 1)] ATUL KUMAR ID-atul.mod@gmail .com mob:09310556579 If n ∈ Q, then (1 + x)n()()()∞+−−+−++.........x!32n1nnx!21nnxn132 = provided x < 1. Note: (i) When the index n is a positive integer the number of terms in the expansion of (1 + x)n is finite i.e., (n + 1) and the coefficient of successive terms are: nC0, nC1, nC2, nC3, ………., nCn(ii) When the index is other than a positive integer such as negative integer or fraction, the number of terms in the expansion of (1 + x). n is infinite and the symbol nCr(iii) Following expansion should be remembered cannot be used to denote the coefficient of the general term. ()1x<. (a) (1 + x)-1 = 1 – x + x2 – x3 + x4 -……. ∞ (b) (1 – x)-1 = 1 + x + x2 + x3 + x4 + …….. ∞ (c) (1 + x)-2 = 1 – 2x + 3x2 – 4x3 + …….. ∞ (d) (1 – x)-2 = 1 + 2x + 3x2 + 4x3(iv) The expansions in ascending powers of x are only valid if x is ‘small’. If x is large i.e., + …….. ∞ 1x> then we may find it convenient to expand in powers of x1, which then will be small. 6. Approximations: (1 + x)n()()().........x3.2.12n1nnx2.11nnxn132−−+−++ = If x < 1, the terms of the above expansion go on decreasing and if x be very small, a stage may be reached when we may neglect the terms containing higher powers of x in the expansion. Thus, if x be so small that its squares and higher powers may be neglected then (1 + x)n = 1 + nx, approximately. This is an approximate value of (1 + x)n7. Exponential Series: . (i) ∞++++=.........!3x!2x!1x1e32x; where x may be any real or complex and nnn11Limite+=∞→ (ii) ∞++++=.........an1!3xan1!2xan1!1x1a3322x where a > 0. Note: a) ∞++++=.........!31!21!111e b) e is an irrational number lying between 2.7 and 2.8. Its value correct upto 10 places of decimal is 2.7182818284. c) ∞++++=+−.............!61!41!2112ee1 d) ∞++++=−−..........!71!51!3112ee1 e) Logarithms to the base ‘e’ are known as the Napierian system, so named after Napier, their inventor. They are also called Natural Logarithm. 8. Logarithmic Series: (i) ()∞+−+−=+.............4x3x2xxx1nl432 where -1 < x ≤ 1. (ii) ()∞+−−−−=−........4x3x2xxx1nl432 where -1 ≤ x < 1.