Show that Rn(x), the lagrange’s form of remainder in the Maclaurin series expansion of , e4s tends to zero as n fi ¥. Hence obtain the Maclaurin’s infinite expansion for e4s .
To show that \( R_n(x) \), the Lagrange form of the remainder in the Maclaurin series expansion of \( e^{4x} \), tends to zero as \( n \) approaches infinity, we can use Taylor's Remainder Theorem. The Lagrange form of the remainder is given by:
\[ R_n(x) = \frac{f^{(n+1)}(c) x^{n+1}}{(n+1)!} \]
where \( c \) is some value between 0 and \( x \). In this case, \( f(x) = e^{4x} \). The \( n \)-th derivative
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Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:
Also check whether the function f is derivable at x = 1.
Show that the function f : R fi R defined byF(X) = 2X + 7 has an inverse by applying the inverse function theorem. Find its inverse also.
Use Cauchy’s mean value theorem to prove that:
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i).
ii).
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i) Set of non-negative integers.
ii) Interval [-1, ¥]
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