in the following exercises, given that 1/(1 - x)=∑(n = 0)^(∞)x^n with convergence in (-1,1), find the power…

Explanation:

Step1: Substitute $u = 4x^{2}$ into $\frac{1}{1 - u}=\sum_{n = 0}^{\infty}u^{n}$

Since $\frac{1}{1 - u}=\sum_{n=0}^{\infty}u^{n}$ for $|u|<1$, substituting $u = 4x^{2}$ gives $\frac{1}{1-4x^{2}}=\sum_{n = 0}^{\infty}(4x^{2})^{n}=\sum_{n=0}^{\infty}4^{n}x^{2n}$.

Step2: Find the interval of convergence

We know that the series $\sum_{n = 0}^{\infty}u^{n}$ converges for $|u|<1$. Substituting back $u = 4x^{2}$, we have $|4x^{2}|<1$. Solving the inequality $4x^{2}<1$, we get $x^{2}<\frac{1}{4}$, so $|x|<\frac{1}{2}$. The interval of convergence is $(-\frac{1}{2},\frac{1}{2})$.

Answer:

The power - series is $\sum_{n = 0}^{\infty}4^{n}x^{2n}$ and the interval of convergence is $(-\frac{1}{2},\frac{1}{2})$