Power Series Representation of ln(1+7x)

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SUMMARY

The power series representation for the function f(x) = ln(1 + 7x) can be derived using the relationship between the function and its derivative. Specifically, the derivative d/dx(ln(1 + 7x)) equals 7/(1 + 7x), which can be expressed as a geometric series. The power series representation is given by the summation from n=1 to infinity of (-7x^n)/n. This method is more efficient than taking multiple derivatives of the original function.

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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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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)?
 
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
 
The connection is that d/dx(ln(1 + 7x) = 7/(1 + 7x), or conversely, that ln(1 + 7x) = \int7/(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.
 

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