The Mystery Behind the Series 1/2! 2/3! 3/4! … and Its Convergence

Mathematics is filled with intricate puzzles, and the series 1/2! 2/3! 3/4! ... is one that leaves many mathematicians intrigued. Let’s explore the series and prove why its sum is exactly 1.

Understanding the Series

The series in question is:

1/2! 2/3! 3/4! ... n/(n 1)! ...

This series is known to be convergent, meaning it has a finite sum. However, the proof of this convergence and its exact sum are not trivial and require a deep understanding of mathematical concepts.

Proving the Convergence of the Series

Let’s start by breaking down the nth term of the series:

For the nth term, we have (frac{n}{(n 1)!}). We can rewrite it by decomposing the factorial term:

[frac{n}{(n 1)!} frac{n}{(n 1) cdot n!} frac{n}{n!} cdot frac{1}{n 1} frac{1}{n!} - frac{1}{(n 1)!}]

Thus, the nth term can be represented as:

[frac{n}{(n 1)!} frac{1}{n!} - frac{1}{(n 1)!}]

Summing the Series

Now, let's sum the series by combining these terms:

[S sum_{n1}^{infty} left( frac{1}{n!} - frac{1}{(n 1)!} right)]

Notice that this is a telescoping series, where most terms cancel out:

[S left( frac{1}{1!} - frac{1}{2!} right) left( frac{1}{2!} - frac{1}{3!} right) left( frac{1}{3!} - frac{1}{4!} right) ldots]

When we sum the series, all intermediate terms cancel out, leaving us with:

[S 1 - lim_{n to infty} frac{1}{(n 1)!}]

Since (frac{1}{(n 1)!}) tends to 0 as (n) approaches infinity, the sum simplifies to:

[S 1 - 0 1]

Using the Exponential Series to Prove the Sum

Another approach to proving the sum is by using the exponential series expansion:

[e^x 1 frac{x}{1!} frac{x^2}{2!} frac{x^3}{3!} ldots]

If we let (x 1), we get:

[e 1 frac{1}{1!} frac{1}{2!} frac{1}{3!} ldots]

Now, consider the series:

[frac{e^x - 1}{x} frac{1}{1!} frac{x}{2!} frac{x^2}{3!} ldots]

Let (x 1), we have:

[frac{e - 1}{1} 1 frac{1}{2!} frac{1}{3!} ldots]

Finally, we can take the derivative of both sides to find:

[1 frac{1}{2!} frac{2}{3!} frac{3}{4!} ldots]

This confirms our earlier result that the sum of the series is indeed 1.

Conclusion

The series 1/2! 2/3! 3/4! ... converges to 1, as shown by both telescoping series and the exponential series approach. Understanding this proof deepens our insight into the beauty and power of mathematical series and their convergence properties.