# Finding a function from its MacLaurin series?

• Xevrex
In summary, the conversation is about a Putnam question involving an integral of two multiplied MacLaurin series. The person is having trouble converting one of the series into a recognizable function and has reduced it to a general form. They are seeking help and have asked for a hint.
Xevrex

## Homework Statement

It's not exactly a specific homework question, but a Putnam one. It's an integral from 0 to inf of two multiplied MacLaurin (as far as I can tell) Series, and I'm trying to figure out how to convert one of them into a recognisable function. I'm really having trouble figuring it out though.

The series itself is $$(x - \frac{x^3}{2} + \frac{x^5}{(2)(4)} - \frac{x^7}{(2)(4)(6)} +\ ...)$$, and I've reduced it to a general form... sort of.

## The Attempt at a Solution

I figured that the series follows the general form of $$\sum_{n=0}^{\infty}\frac{(-1)^{n-1}(2n-1)!x^{2n-1}}{(2n-1)!}$$. It looks reminiscent of something like sin x, but I have no clue what deviation from that function would have to occur to produce that series.

By the way, I haven't formally learned Taylor/MacLaurin series, but I understand the general concepts of them--but if the method I'm asking for is generally taught within the unit, then I'm dreadfully sorry for wasting everyone's time. Every internet search I've done so far has yet to turn up anything, so...

Welcome to PF!

Hi Xevrex!Welcome to PF!

Hint: try putting y = x2/2

## 1. What is a MacLaurin series?

A MacLaurin series is a special type of power series representation of a function, where the center of expansion is at x = 0. It is named after mathematician Colin MacLaurin.

## 2. How do you find a function from its MacLaurin series?

To find a function from its MacLaurin series, we use the process of term-by-term differentiation and integration. This involves finding the coefficients of the series and then using them to construct the original function.

## 3. What is the significance of finding a function from its MacLaurin series?

Finding a function from its MacLaurin series allows us to represent a complicated function as an infinite polynomial, making it easier to work with and manipulate. It also helps us approximate the value of the function at a specific point.

## 4. Can any function be represented by a MacLaurin series?

Yes, any function that is infinitely differentiable at x = 0 can be represented by a MacLaurin series. However, the series may not converge for all values of x, so it is important to check for convergence before using the series to approximate the function.

## 5. Are there any applications of finding a function from its MacLaurin series?

Yes, finding a function from its MacLaurin series is useful in various areas of mathematics, physics, and engineering. It is used to solve differential equations, approximate complex functions, and analyze the behavior of systems in the real world.

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