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(Solved): Prove that 4ex is equal to the sum of its Maclaurin series. Solution If f(x)=4ex, then f(n+1)(x)= f ...
Prove that 4ex is equal to the sum of its Maclaurin series. Solution If f(x)=4ex, then f(n+1)(x)= for all n. If d is any positive number and ∣x∣≤d,then∣∣f(n+1)(x)∣∣= So Taylor's Inequality, with a=0 and M=4ed, says that ∣Rn(x)∣≤(n+1)!∣x∣n+1 for ∣x∣≤d. Notice that the same constant M=4ed works for every value of n. But, from this equation, we have limn→∞(n+1)!−4ed∣x∣n+1=4edlimn→∞(n+1)!∣x∣n+1= It follows from the Squeeze Theorem that limn→∞∣Rn(x)∣=0 and therefore limn→∞Rn(x)= for ail values of x. By this theorem, Ae ex is equal to the sum of its Maclaurin series, that is, 4ex=∑n=0∞n!4xn for all x.