use the geometric series f(x) = 1 / (1 - x) = ∑(k = 0 to ∞) x^k, for |x| < 1, to find the power - series…

use the geometric series f(x) = 1 / (1 - x) = ∑(k = 0 to ∞) x^k, for |x| < 1, to find the power - series representation for the following function (centered at 0). give the interval of convergence of the new series. f(4x) = 1 / (1 - 4x) which of the following is the power - series representation for f(4x)? oa. ∑(k = 0 to ∞) x^(4k) ob. ∑(k = 0 to ∞) 4x^k oc. ∑(k = 0 to ∞) 1 / (1 - (4x)^k) od. ∑(k = 0 to ∞) (4x)^k

use the geometric series f(x) = 1 / (1 - x) = ∑(k = 0 to ∞) x^k, for |x| < 1, to find the power - series representation for the following function (centered at 0). give the interval of convergence of the new series. f(4x) = 1 / (1 - 4x) which of the following is the power - series representation for f(4x)? oa. ∑(k = 0 to ∞) x^(4k) ob. ∑(k = 0 to ∞) 4x^k oc. ∑(k = 0 to ∞) 1 / (1 - (4x)^k) od. ∑(k = 0 to ∞) (4x)^k

Answer

Explanation:

Step1: Recall geometric - series formula

The geometric series is $\frac{1}{1 - x}=\sum_{k = 0}^{\infty}x^{k}$, for $|x|\lt1$.

Step2: Substitute $4x$ for $x$

We want to find the power - series for $\frac{1}{1 - 4x}$. Let $u = 4x$. Then, by the geometric series formula, $\frac{1}{1 - 4x}=\sum_{k = 0}^{\infty}(4x)^{k}$, for $|4x|\lt1$.

Step3: Determine the interval of convergence

Solve the inequality $|4x|\lt1$. Divide both sides by 4: $|x|\lt\frac{1}{4}$. The interval of convergence is $(-\frac{1}{4},\frac{1}{4})$.

Answer:

D. $\sum_{k = 0}^{\infty}(4x)^{k}$