What is the mistake in this attempt to prove \sum {\frac{x^n}{n}} = -ln(1-x)?

In summary, the conversation discusses an attempt to show that a specific equation is equal to the negative natural logarithm of (1-x). The mistake is found in the integration and the conversation explains the correct way to integrate the function to reach the desired result.
  • #1
talolard
125
0

Homework Statement



I am trying to show that [tex]\sum{ \frac{x^n}{n}} = -ln(1-x)[/tex]
But I am doing something wrong and I can't find my mistake.
Please find my mistake and let me know what it is.
Thanks

The Attempt at a Solution


set [tex] f(x)=\sum {\frac{x^n}{n}} [/tex]
then [tex] f'(x)= \sum {x^n-1} [/tex]
so [tex] xf'(x)=\sum{x^n} = \frac {1}{1-x} [/tex]
which means that [tex] f'(x)=\frac {1}{x} - \frac{1}{1-x} [/tex]
integrtang we get [tex] f(x)=ln|x|-ln|1-x| [/tex]
which is bad






 
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  • #2
The derivative of the function is:

f'(x)=Sum[x^(n-1),{n,1,Infinity}]

which is the same as:

f'(x) = Sum[x^(n),{n,0,Infinity}] = 1/(1-x)

Integrating:

Sum[x^(n+1)/(n+1),{n,0,Infinity}] = - ln(1-x)

which is the same as:

Sum[x^(n)/n,{n,1,Infinity}] = -ln(1-x).

The trick lies in repositioning the starting index.
 

1. What is a simple generating function?

A simple generating function is a mathematical tool used to represent a sequence of numbers in a concise and systematic way. It is a power series in which each term represents a coefficient of the sequence.

2. How is a simple generating function different from other generating functions?

A simple generating function only includes the coefficients of a sequence, whereas other generating functions may also include additional variables and parameters. This makes simple generating functions easier to work with and understand.

3. What are some applications of simple generating functions?

Simple generating functions are commonly used in combinatorics, probability, and number theory to solve problems involving counting and recurrence relations. They can also be used in physics and engineering to model and analyze systems with discrete states.

4. Can a simple generating function be used for infinite sequences?

Yes, simple generating functions can represent both finite and infinite sequences. However, for infinite sequences, the function may only converge for certain values of the variable.

5. How do I find the coefficient of a specific term in a simple generating function?

To find the coefficient of a term in a simple generating function, you can use the formula for the coefficient of a power series. This involves taking the derivative of the function and evaluating it at the desired value of the variable. Alternatively, you can also use the Cauchy Integral Formula to calculate the coefficient.

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