Express this series in terms of the given series A(x)

In summary, we are given the function A(x) = a0 + a1x + a2x2 + a3x3 + ... = Ʃanxn and asked to express E(x) = a0 + a2x2 + a4x4 + ... = Ʃa2nx2n and O(x) = a1x + a3x3 + a5x5 + ... in terms of A(x). Using the fact that A(-x) = a0 - a1x + a2x2 - a3x3 + ..., we can see that adding A(x) and A(-x) will result in 2*E(x) while subtracting A(x) from
  • #1
ptolema
83
0

Homework Statement



Let A(x) = a0 + a1x + a2x2 + a3x3 + ... = Ʃanxn.

Express E(x) = a0 + a2x2 + a4x4 + ... = Ʃa2nx2n.
Do the same for O(x) = a1x + a3x3 + a5x5 + ...

Homework Equations



A(x) = a0 + a1x + a2x2 + a3x3 + ... = Ʃanxn

The Attempt at a Solution


So I tried using A(x2) as a starting point, but got stuck on how to get the coefficients of A(x2) = a0 + a1x2 + a2x4 + a3x6 + ... to match those of E(x).
How would I begin expressing E(x) and O(x) in terms of A(x)? Could there be something I could do with sin or cos?
 
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  • #2
Think about A(-x). What is its series?
 
  • #3
awkward said:
Think about A(-x). What is its series?

A(-x) = a0 - a1x + a2x2 - a3x3 + ...
I see now. So when I add A(x)+A(-x), I get 2*E(x). Similarly, A(-x)-A(x) = -2*O(x). Thanks, your hint was a huge help!
 

FAQ: Express this series in terms of the given series A(x)

What is the formula for expressing this series in terms of the given series A(x)?

The formula for expressing a series in terms of another series, A(x), is to use the substitution method. This involves replacing each term in the series with the corresponding term in A(x). For example, if the series is 1, 4, 9, 16 and A(x) is x, then the series in terms of A(x) would be x, 2x, 3x, 4x.

Can any series be expressed in terms of the given series A(x)?

Yes, any series can be expressed in terms of the given series A(x) as long as the terms in the series can be related to the terms in A(x) through a formula or pattern. This means that A(x) can act as a universal series for expressing other series.

What are some common examples of using A(x) to express a series?

Some common examples of using A(x) to express a series include geometric series, arithmetic series, and power series. For example, the series 2, 8, 18, 32 can be expressed in terms of A(x) as 2x^2, 4x^3, 6x^4, 8x^5.

What is the purpose of expressing a series in terms of A(x)?

The purpose of expressing a series in terms of A(x) is to simplify the series and make it easier to understand and work with. It can also help identify patterns and relationships between different series.

Is there a specific method for finding the series in terms of A(x)?

Yes, there are various methods for finding the series in terms of A(x), including the substitution method, the partial fraction method, and the generating function method. The method used will depend on the specific series and the resources available.

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