Help Needed: Checking & Solving Power Series for f(x) = x*ln(1+x)

In summary, the conversation discusses finding power series representations for two functions: f(x) = ln(1+x) and f(x) = x*ln(1+x). The first part involves finding a power series representation for ln(1+x) by differentiating and integrating the power series representation for \frac{1}{1+x}. The second part involves combining two power series to find a representation for x*ln(1+x).
  • #1
tandoorichicken
245
0
Two parts to this problem. On the first part I need someone to check my work, and I need help on solving the second part.

(a) Find a power series representation for f(x) = ln(1+x).
[tex] \frac{df}{dx} = \frac{1}{1+x} = \frac{1}{1-(-x)} = 1-x+x^2-x^3+x^4-...[/tex]
[tex] \int_{n} \frac{1}{1+x} = \int_{n} (1-x+x^2-x^3+...)\dx = x-\frac{x^2}{2}+\frac{x^3}{3}-...+C = \sum^{\infty}_{n=0} \frac{(-x)^{n+1}}{n+1} +C [/tex]
sub x=0 in original equation: ln(1+0) = ln(1) = 0 = C.
[tex]\ln(1+x) = \sum^{\infty}_{n=0} \frac{(-x)^{n+1}}{n+1}[/tex]
with radius of convergence = 1.

(b) Find a power series representation for f(x) = x*ln(1+x).
If this function is differentiated, you get
[tex] \ln(1+x) + \frac{x}{x+1}[/tex]
which is the same as a sum of power series
[tex] \sum^{\infty}_{n=0} \frac{(-x)^{n+1}}{n+1} + \sum^{\infty}_{n=0} (-x)^{n+1} [/tex]
have I gone too far? or where do I go from here?
I know I will eventually have to integrate back to get the series for the original function.
 
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  • #2
The first series looks right except for a sign, but why don't you just replace n+1 by n and run the index from 1 to infinity? For the second one, you can just multiply each term from the expansion for ln(1+x) by x.
 
Last edited:
  • #3
(a) looks fine (except for the sign), though it's usually good form to reindex the power series, and rearrange the terms, so that it looks like

[tex]
\sum_{n = ?}^{\infty} (\mathrm{something}) x^n
[/tex]

(b) You don't have to do any differentiation at all... :smile: but you could do it that way if you really want to, by combining the two sums.
 

Related to Help Needed: Checking & Solving Power Series for f(x) = x*ln(1+x)

1. What is a power series and how is it used in mathematics?

A power series is a mathematical representation of a function as an infinite sum of powers of a variable. It is used in calculus and other areas of mathematics to approximate functions and solve equations.

2. How do you check if a power series is convergent or divergent?

To check if a power series is convergent or divergent, you can use a variety of tests such as the ratio test, root test, or comparison test. These tests compare the series to known convergent or divergent series and determine its behavior.

3. What is the function f(x) = x*ln(1+x) and why is it important?

The function f(x) = x*ln(1+x) is a power series that represents the natural logarithm function ln(x). It is important in mathematics as it is used to solve various problems in calculus, differential equations, and other areas of study.

4. How do you solve a power series for a given function?

To solve a power series for a given function, you can use various techniques such as integration, substitution, and manipulation of the series. It is important to carefully evaluate the series and check for convergence before using it to approximate a function.

5. Can power series be used to solve real-world problems?

Yes, power series can be used to solve real-world problems in fields such as physics, engineering, and economics. They are often used to approximate complicated functions and make predictions about the behavior of a system.

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