What is the interval of convergence of $\infty \sum 1 {(- 2)}^{n} (n + 1) {(x - 1)}^{n}$ $sum_1^oo (-2)^n(n+1)(x-1)^n$ ?

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Answer 1 · 2016-01-28T23:14:30

$(\frac{1}{2}, \frac{3}{2}]$ $(1/2,3/2]$

Use the ratio test.

The infinite sum $\infty \sum a_{n}$ $sum^ooa_n$ converges when

$lim_(nrarroo)abs((a_(n+1))/(a_n))<1$

This gives

$lim_(nrarroo)abs(((-2)^(n+1)(n+2)(x-1)^(n+1))/((-2)^n(n+1)(x-1)^n))<1$

Simplified:

$lim_(nrarroo)abs((-2(n+1)(x-1))/(n+2))<1$

Evaluating the limit yields

$abs(-2(x-1))<1$

Resulting in the inequality

$1/2 < x < 3/2$

We now have to plug in $x=1/2$ and $x=3/2$ to see if the sum converges at the endpoints.

Plugging in $x=1/2$ does not converge, but $x=3/2$ does.

So, the answer is

$1/2 < x <=3/2$ or $(1/2,3/2]$

What is the interval of convergence of sum_1^oo (-2)^n(n+1)(x-1)^n ∞∑1(−2)n(n+1)(x−1)nsum_1^oo (-2)^n(n+1)(x-1)^n ?