Animated Solution for Mathematics - Sequence and Series: The sum of series $\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\dots$ upto infinity is

The series we are evaluating is given by:

The presence of factorials in the denominator is a significant clue. In calculus, whenever you encounter $n!$ in the denominator, your mind should immediately associate it with the Maclaurin expansion of the exponential function.

The standard expansion is defined as:

This identity serves as the bedrock of our solution.

Observe the signs in our target series; they alternate between positive and negative. Since the standard expansion of $e^x$ contains only positive terms, we must force the signs to flip.

We achieve this by substituting a negative value for $x$. Specifically, if we set $x=-1$, the term $x^n$ becomes $(-1{)}^n$.

Substituting $x=-1$ into the expansion, we obtain:

Simplifying the powers of $-1$, we get:

Look closely at the first two terms: $1-1$. These terms sum to zero and effectively vanish.

This leaves us with exactly the series we were asked to evaluate:

Since $e^{-1}=\frac{1}{e}$, the final value of the series is:

You have successfully solved this infinite series by recognizing the hidden structure of the exponential function. Remember: when you see factorials, think exponentials; when you see alternating signs, think negative bases.