For k 0,calculate the value of sum_{x=0}^{i | vedclass

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(D) Given the expression: $\sum_{x=0}^{\infty} \frac{k^x}{x !} \lim _{n}$ ${\rightarrow \infty} \frac{n !}{(n-x) !}\left(1-\frac{k}{n}\right)^{n-x}\le

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$1$

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(D) Given the expression: $\sum_{x=0}^{\infty} \frac{k^x}{x !} \lim _{n}$ ${\rightarrow \infty} \frac{n !}{(n-x) !}\left(1-\frac{k}{n}\right)^{n-x}\left(\frac{1}{n}\right)^x$ $= \lim _{n}$ ${\rightarrow \infty} \sum_{x=0}^n \frac{n !}{x !(n-x) !}\left(1-\frac{k}{n}\right)^{n-x}\left(\frac{k}{n}\right)^x$ $= \lim _{n \rightarrow \infty} \sum_{x=0}^n { }^n C_x \left(1-\frac{k}{n}\right)^{n-x}\left(\frac{k}{n}\right)^x$ Using the Binomial Theorem,$\sum_{x=0}^n { }^n C_x a^{n-x} b^x = (a+b)^n$: $= \lim _{n \rightarrow \infty} \left(1-\frac{k}{n} + \frac{k}{n}\right)^n$ $= \lim _{n \rightarrow \infty} (1)^n = 1$ Thus,the correct option is $D$.

For $k > 0$,calculate the value of $\sum_{x=0}^{\infty} \frac{k^x}{x !} \lim _{n}$ ${\rightarrow \infty} \frac{n !}{(n-x) !}\left(1-\frac{k}{n}\right)^{n-x}\left(\frac{1}{n}\right)^x$.

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