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E25) Find the coefficient of x3 in Taylor's series around zero for the function sin-' x.
Some General Remarks on Taylor's and Maclaurin's Series Though we have obtained infinite series for many functions, it is necessary to give a note of caution. These Infinite series need not be valid for all values of x, and as such, these have to be used with care. In the course on real analysis, you will be able to study the conditions under which these series are valid.
1.6
SUMMARY
In this unit, we have 1)
introduced higher order derivatives,
2)
derived a formula (Leibniz's Theorem) for the nIhderivative of a product of two functions. (uv), = C(n, 0) u, v + C(n, 1) u, v, + C(n, 2) u, v2+ ........+ C(n, n) u v,.
3)
written Taylor's series around zeroiMaclaurin's series of a number of functions by using the formula
1.7
SOLUTIONS AND ANSWERS
E l ) a) 6x E2) a) 11
k
#
0, since
k=O*O=qJ?;, which is impossible.
JZ
* f12)(x) =- k2sin kx * c2)( d 6 ) =- k2sin knl6
E4) a) f(x)= sin kx
Now,-k2sinkd6=2&*sinkd6=-2&lk2 Since- 1 ~ s i n k ~ l 6 < O , - n < k x 1 6 < 0 * k = - 1 or.-2or-3or-4or-5 Out of these, k = - 2 is the value which satisfies sin k A 16 = - 21 & lk2