Power Series Representation of ln(1+7x)

In summary, there are two ways to find a power series representation for a function like f(x)= ln(1+7x). One method involves taking derivatives of the function, while the other method is quicker and involves finding a power series for 7/(1 + 7x) = 1/(x + 1/7). The connection between ln(1 + 7x) and 1/(x + 1/7) is that their derivatives are equal to each other, or ln(1 + 7x) can be represented as an integral of 7/(1 + 7x) dx. However, the correctness of the result may be questionable.
  • #1
sportlover36
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how can i find a power series representation for a function like f(x)= ln(1+7x)?
 
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  • #2
sportlover36 said:
how can i find a power series representation for a function like f(x)= ln(1+7x)?
There are at least two ways. One is very mechanical and involves taking the derivatives of your function. The other is probably a lot quicker. Can you find a power series for 7/(1 + 7x) = 1/(x + 1/7)? What's the connection between ln(1 + 7x) and 1/(x + 1/7)?
 
  • #3
Im not really sure what the connection is but i got this at the answer...the sumation from n=1 to infinity -7x^n/n
 
  • #4
The connection is that d/dx(ln(1 + 7x) = 7/(1 + 7x), or conversely, that ln(1 + 7x) = [itex]\int[/itex]7/(1 + 7x) dx.

If you didn't understand that connection, how did you get the result that you got? BTW, I'm not sure that your result is right.
 

1. What is the power series representation of ln(1+7x)?

The power series representation of ln(1+7x) is n=1 (-1)n+1 (7x)n/n.

2. How is the power series derived for ln(1+7x)?

The power series for ln(1+7x) is derived using the Maclaurin series expansion, which is a special case of the Taylor series expansion. This involves finding the derivatives of ln(1+7x) and evaluating them at x = 0.

3. What is the interval of convergence for the power series representation of ln(1+7x)?

The interval of convergence for the power series representation of ln(1+7x) is -1/7 < x < 1/7. This means that the series will only converge for values of x within this interval.

4. How accurate is the power series representation of ln(1+7x)?

The accuracy of the power series representation of ln(1+7x) depends on how many terms are used in the series. The more terms included, the closer the approximation will be to the actual value of ln(1+7x).

5. What are the applications of the power series representation of ln(1+7x)?

The power series representation of ln(1+7x) can be used to approximate ln(1+7x) in situations where it is difficult to calculate the value directly. It is also used in mathematical and scientific computations, such as in finding the area under a curve or solving differential equations.

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