Converting 1/(1+7x)^2 into a Power Series

In summary, to find the power series for the function f(x) = 1/(1+7x)^2 from n=1 to infinity, you can use the general formula for series and take the derivative of the summation. Using this method, you should get the answer of (-1)^n * 7 * 7^n * x^n when n=1 to infinity. It is important to note that you will need to find a way to incorporate the -7 into the summation.
  • #1
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Homework Statement


i need to make the function 1/(1+7x)^2 into a power series that goes from n=1 to infinity. I know that i have to get the answer through differentiating because the for the previous problem, i found that the function 7/(1+7x) resulted in ( -1 )^n* 7*7^n*x^n when n=0 to infinity

Homework Equations


The Attempt at a Solution


I know that as 1/(1-x) equals x^n and 1/(1-x)^2 equals n*x^(n-1) when n begins at 1 so i tried to use that. my previous try was (1)^n*(-1)*x^(n-1)*7^(2n-1)*(n). I'm on my last try and I really need help!
 
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  • #2
f = 1/(1+7x)
f' = -7/(1 + 7x)^2

General formula for series:
1/(1-x) = sum[0-->infinite](x^n)

So, 1/(1-(-7x)) = sum[0-->infinite]((-7x)^n) = sum[0-->infinite]((-1)^n(7x)^n)

Now, just take the derivative of the summation, and you should get your answer.
Also, you'll have to find a way to get that -7 into the summ ;).
 
  • #3
thank you so much! that was incredibly helpful
 

1. What is a power series?

A power series is an infinite sum of terms that involve a variable raised to different powers. It is commonly used in mathematics and physics to approximate functions.

2. When do I need to use a power series?

A power series is useful when working with functions that cannot be easily integrated or differentiated, or when a function is too complicated to work with directly. It can also be used to approximate a function at a specific point or to find the behavior of a function at infinity.

3. How do I find the coefficients of a power series?

The coefficients of a power series can be found by using the Taylor series or Maclaurin series. These are specific types of power series that are used to approximate a function at a specific point or at x=0, respectively.

4. Can a power series represent any function?

No, not all functions can be represented by a power series. A function must be analytic (meaning it can be represented by a power series) in order for a power series to accurately approximate it.

5. Are there any special cases when using a power series?

Yes, there are a few special cases to keep in mind when using a power series. For example, when the function being approximated has a singularity (such as a vertical asymptote), the power series may not converge. Also, when using a power series to approximate a periodic function, the power series may only converge within a specific interval.

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