# Power Series Representation of ln(1+7x)

• sportlover36
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.
sportlover36
how can i find a power series representation for a function like f(x)= ln(1+7x)?

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) = $\int$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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