Series Expansion: Proving ln(1+x)/x

In summary, the given function can be rewritten as a geometric series with alternating signs. By incorporating (1)n into the series, the ratio of consecutive terms becomes independent of n, making it a geometric series. This allows us to easily find the integral and ultimately arrive at the desired result of ln(1+x)/x.
  • #1
thercias
62
0

Homework Statement


Show that 1- x/2 + x^2/3 - x^3/4 + x^4/5... (-1)^n (x^n)/(n+1) = ln(1+x)/x
with |x| < 1

Homework Equations




The Attempt at a Solution


finding derivative of the function multiplied by x
d/dx(xS(x))
= 1 -x + x^2 - x^3 + x^4 - x^5 +...

absolute value of this function = 1 + x + x^2 + x^3 + x^4... = 1/1-x ( geometric series)
the integral of that gives us -ln(1-x) + c

drawing a blank now. not really sure where to go.
 
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  • #2
thercias said:
= 1 -x + x^2 - x^3 + x^4 - x^5 +...

absolute value of this function = 1 + x + x^2 + x^3 + x^4... = 1/1-x ( geometric series)
No it isn't. But it is a geometric series, so you can sum it quite easily.
 
  • #3
ok the absolute value part is wrong, but what do you mean i can sum it up easily? i don't get it. is the integral still -ln(1-x) + c..? i know the geometric series for 1 + x + x^2 + ... = 1/1-x, but how does that change with alternating signs? is it going to be (-1)^n*(1/1-x)? so the integral is (-1)^n*-ln(1-x)? how do i go from that to ln(1+x)/x?
 
Last edited:
  • #4
An alternating series involves (1)n which can easily be incorported into the geometric series.
 
  • #5
HallsofIvy said:
An alternating series involves (1)n which can easily be incorported into the geometric series.
I think you meant (-1)n.
Thercias, what is the ratio of consecutive terms in your alternating series? If it's independent of n then it's a geometric series.
 

1. What is series expansion?

Series expansion is a method used in mathematics to express a function as an infinite sum of simpler functions. It is particularly useful in proving mathematical statements and solving equations.

2. How is series expansion used in proving ln(1+x)/x?

In proving ln(1+x)/x, series expansion is used to express the function as an infinite sum of simpler functions. This allows us to manipulate the function and prove its properties using known identities and theorems.

3. Why is proving ln(1+x)/x important?

Proving ln(1+x)/x is important because it is a fundamental result in calculus and is used in many other mathematical concepts and applications. Additionally, it helps us understand the properties of logarithmic and exponential functions.

4. What is the significance of using ln(1+x)/x in series expansion?

Using ln(1+x)/x in series expansion allows us to prove the properties of logarithmic and exponential functions. It also helps us approximate the values of these functions for various inputs, which is useful in many real-world scenarios.

5. Are there any limitations to using series expansion in proving ln(1+x)/x?

Yes, there are limitations to using series expansion in proving ln(1+x)/x. One limitation is that the series may only converge for certain values of x, so the result may not be applicable for all inputs. Additionally, the process of finding the series expansion can be complex and time-consuming.

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