Identifying the Function via Maclaurin Series Expansion: sum_{n1}^infty x^n /n2! and ex

One of the fundamental tools in calculus is the Maclaurin series, which allows us to express functions as infinite series. In this article, we will explore how to identify the function whose Maclaurin series is given by sum_{n1}^infty x^n /n2!.

Introduction

The Maclaurin series of a function f(x) is a Taylor series expansion centered at x 0. It is expressed as:

f(x) sum_{n0}^infty frac{f^{(n)}(0)}{n!} x^n

where f^{(n)}(0) denotes the nth derivative of f(x) evaluated at x 0.

The Given Series

The given series is:

sum_{n1}^infty frac{x^n}{n2!}

Our goal is to identify the function f(x) such that its Maclaurin series matches the given series.

Series Manipulation

Let's start by rewriting the given series in a more recognizable form. Notice that the series starts at n 1, which means we can relate it to the known Maclaurin series for the exponential function e^x:

e^x sum_{n0}^infty frac{x^n}{n!}

The given series can be expressed in terms of the series for e^x by making some substitutions.

Step-by-Step Derivation

First, let's manipulate the given series:

sum_{n1}^infty frac{x^n}{n2!} sum_{n3}^infty frac{x^{n-2}}{n!}

Now, let m n - 2, then we have:

sum_{n3}^infty frac{x^{n-2}}{n!} sum_{m1}^infty frac{x^m}{(m 2)!}

To further simplify, we can rewrite:

sum_{m1}^infty frac{x^m}{(m 2)!} frac{1}{x^2} sum_{m1}^infty frac{x^m}{m!}

Therefore, the given series can be expressed as:

sum_{n1}^infty frac{x^n}{n2!} frac{1}{x^2} sum_{m1}^infty frac{x^m}{m!}

Recall that the Maclaurin series for e^x is:

e^x sum_{n0}^infty frac{x^n}{n!} 1 x sum_{n2}^infty frac{x^n}{n!}

Therefore:

sum_{n2}^infty frac{x^n}{n!} e^x - 1 - x

Combining these results, we get:

sum_{n1}^infty frac{x^n}{n2!} frac{1}{x^2} (e^x - 1 - x)

The Resulting Function

The function whose Maclaurin series is given by the original series is:

frac{e^x - 1 - x}{x^2}

Thus, we have identified the function:

f(x) frac{e^x - 1 - x}{x^2}

Conclusion

In this article, we have demonstrated how to identify a function via its Maclaurin series expansion. Specifically, we showed that the function whose Maclaurin series is given by

sum_{n1}^infty frac{x^n}{n2!}

is

frac{e^x - 1 - x}{x^2}

This process involves manipulating the series and using known Maclaurin series expansions to simplify and identify the function.

Key Points

Maclaurin series expansion of a function Identifying functions using series Manipulation of series to find known functions

For further reading and practice, you can explore more examples involving series expansions and their applications in calculus and advanced mathematics.