Question

find a power series representation for the function. (give your power series representation centered at x = 0.) f(x) = 6 7 − x

Answer 1

To find the power series representation of f(x) = 6/(7-x), we start with the power series for 1/(1-x), multiply by 6, and adjust for the negative sign in the denominator, resulting in \(\sum_{n=0}^{\infty} (-1)^n 6x^n\).

Step-by-step explanation:

The question is asking to find a power series representation of the function f(x) = \(\frac{6}{7-x}\), centered at x = 0. To find this representation, one can start by finding the power series of \(\frac{1}{1-x}\), then multiply this series by 6, and finally replace x with \(-x\) to adjust for the minus sign in the denominator of the original function. The power series of

`\((1)/(1-x)\) is:`

`\(1 + x + x^2 + x^3 + \cdots\)`

So for our function, we multiply the series by 6:

`\(6(1 + (-x) + (-x)^2 + (-x)^3 + \cdots)\)`

This gives us:

`\(6 - 6x + 6x^2 - 6x^3 + \cdots\)`

Therefore, the power series representation of f(x) = \(\frac{6}{7-x}\), centered at x = 0 is:

`\(\sum_(n=0)^(\infty) (-1)^n 6x^n\)`

Answer 2

Final Answer:

The power series representation for f(x) = 6/(7 - x), centered at x = 0, is:

6 + 6x + 36x^2 + 216x^3 + ... + 6^n * n! * x^n / (7^n + 1) + ...

Step-by-step explanation:

We can find the power series representation of f(x) using the geometric series formula:

1 / (1 - z) = 1 + z + z^2 + z^3 + ...

This formula is valid for |z| < 1. We can rewrite f(x) using a manipulation:

f(x) = 6 / (7 - x) = 6 * (1 / (1 - (x/7)))

Now, if |x/7| < 1, we can substitute the geometric series formula into the expression:

f(x) = 6 * (1 + (x/7) + (x/7)^2 + (x/7)^3 + ...)

Simplifying the terms and multiplying by 6:

6 + 6x + 36x^2 + 216x^3 + ... + 6^n * n! * x^n / (7^n + 1) + ...

This is the power series representation of f(x) centered at x = 0. Note that the series converges when |x/7| < 1, which means the interval of convergence is (-7, 7).