Question1. [-/2 Points] DETAILS MY NOTES SCALCET9 11.9.003.MI.
Find a power series representation for the function. (Center your power series representation at $x=0$.) $$\begin{array}{r} f(x)=\frac{1}{7+x} \\ f(x)=\sum_{n=0}^{\infty}(\square) \end{array}$$
Determine the interval of convergence. (Enter your answer using interval notation.) $\square$ Submit Answer

Studdy Solution
Determine the interval of convergence. For a geometric series $\sum_{n=0}^{\infty} r^n$, the series converges if $|r| < 1$.
For our series, $r = -\frac{x}{7}$, so we require:
$$\left| -\frac{x}{7} \right| < 1$$
This simplifies to:
$$\left| x \right| < 7$$
Therefore, the interval of convergence is:
$$(-7, 7)$$
The power series representation is:
$$f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{7^{n+1}}$$
And the interval of convergence is:
$$(-7, 7)$$