Show that  the lagrange’s form of remainder in the Maclaurin series expansion of  , tends to zero as n → ∞.Hence obtain the Maclaurin’s infinite expansion for 

We are tasked with showing that the Lagrange remainder Rn(x)R_n(x) in the Maclaurin series expansion of e4xe^{4x} tends to zero as n→∞n \to \infty, and from this, obtaining the infinite Maclaurin expansion for e4xe^{4x}.

Step 1: General Form of the Lagrange Remainder

The Lagrange remainder for the Maclaurin series expansion of a function f(x)f(x) is given by:

Rn(x)=f(n+1)(c)(n+1)!xn+1R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}

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