1. -/1.66 points details my notes scalcet9 11.8.0 find the radius of convergence, r, of the series. ∑n = 1…

Explanation:

Step1: Apply ratio - test formula

Let $a_n = 7(-1)^nnx^n$. Then $a_{n + 1}=7(-1)^{n+1}(n + 1)x^{n+1}$. Calculate $\lim_{n\rightarrow\infty}\left|\frac{a_{n + 1}}{a_n}\right|=\lim_{n\rightarrow\infty}\left|\frac{7(-1)^{n + 1}(n + 1)x^{n+1}}{7(-1)^nnx^n}\right|$.

Step2: Simplify the ratio

$\lim_{n\rightarrow\infty}\left|\frac{a_{n + 1}}{a_n}\right|=\lim_{n\rightarrow\infty}\left|\frac{(-1)(n + 1)x}{n}\right|=\lim_{n\rightarrow\infty}\left|\left(1+\frac{1}{n}\right)x\right|=|x|$.

Step3: Find the radius of convergence

For convergence, $\lim_{n\rightarrow\infty}\left|\frac{a_{n + 1}}{a_n}\right|<1$. So $|x|<1$. The radius of convergence $R = 1$.

Step4: Test the endpoints

When $x = 1$, the series becomes $\sum_{n = 1}^{\infty}7(-1)^nn(1)^n=\sum_{n = 1}^{\infty}7(-1)^nn$, which diverges by the divergence - test (since $\lim_{n\rightarrow\infty}7(-1)^nn$ does not exist). When $x=-1$, the series becomes $\sum_{n = 1}^{\infty}7(-1)^nn(-1)^n=\sum_{n = 1}^{\infty}7n$, which also diverges by the divergence - test ($\lim_{n\rightarrow\infty}7n=\infty$).

Answer:

$R = 1$ $I=(-1,1)$