2. the sum of the series 1 + 4 + 4²/2! + 4³/3! + ⋯ + 4ⁿ/n! + ⋯ is
2. the sum of the series 1 + 4 + 4²/2! + 4³/3! + ⋯ + 4ⁿ/n! + ⋯ is
Answer
Explanation:
Step1: Recall the power - series formula
The power - series of the exponential function is given by $e^{x}=\sum_{n = 0}^{\infty}\frac{x^{n}}{n!}=1 + x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots+\frac{x^{n}}{n!}+\cdots$.
Step2: Identify the value of x
In the given series $1 + 4+\frac{4^{2}}{2!}+\frac{4^{3}}{3!}+\cdots+\frac{4^{n}}{n!}+\cdots$, we can see that $x = 4$.
Answer:
$e^{4}$