# How do you find a power series representation for # (1+x)/((1-x)^2)#?

Fair warning---I expect this to be a long answer!

I got

So, something that I believe you have already been taught is that:

and that:

Analogizing from (1):

Remember these four relationships, because we will be referring back to them.

First, notice how you can rewrite this as:

Already you may see how things could unfold. Making use of (0) in conjunction with (2), we get:

Now, making use of (1), we get:

This becomes the left half that we are looking for. Now for the right half. Using (3), we get:

Next, we can add them together (like-terms with like-terms) according to (4):

The negative sign carried through all our operations.

Wolfram Alpha agrees with this answer.

By signing up, you agree to our Terms of Service and Privacy Policy

To find a power series representation for ( \frac{{1+x}}{{(1-x)^2}} ), we can start with the geometric series formula ( \frac{1}{{1-x}} = \sum_{n=0}^{\infty} x^n ). Taking the derivative of both sides, we get ( \frac{1}{{(1-x)^2}} = \sum_{n=1}^{\infty} nx^{n-1} ). Then, multiplying by ( 1+x ), we have ( \frac{{1+x}}{{(1-x)^2}} = \sum_{n=1}^{\infty} nx^{n-1} (1+x) ). Finally, we simplify this expression to get ( \frac{{1+x}}{{(1-x)^2}} = \sum_{n=1}^{\infty} (n+1)x^n + \sum_{n=1}^{\infty} nx^{n-1} ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the interval of convergence #Sigma x^n/n^2# from #n=[1,oo)#?
- How do you find the first two nonzero terms in Maclaurin's Formula and use it to approximate #f(1/3)# given #f(x)=int_0^x sin(t^2) dt# ?
- How do you find the taylor series for #(1/(1-x^2))# centered around #a = 3#?
- How do you find the Maclaurin series for # f(x)=(9x^2)e^(−7x)#?
- How do you find the Maclaurin Series for #f(x)= x / (1-x^4)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7