Question

The limit of $\sum _{n-1}^{1000}(-1{)}^n{x}^n$ as $x\to \infty$

• A. does not exist
• B. exists and equals to 0
• C. exists and approaches to $+\propto$
• D. exists and approaches $-\infty$

Answer 1

Answer: (C)

$\underset{x\to \infty }{\lim }\sum _{n=1}^{1000}(-1{)}^n{x}^n$
$=\underset{x\to \infty }{\lim }\left\{-x+x^2-x^3+x^4+\dots +x^{1000}\right\}$
$=\underset{x\to \infty }{\lim }(-x)\cdot \left\{\frac{(-x{)}^{1000}-1}{(-x-1)}\right\}=\underset{x\to \infty }{\lim }\frac{x^{1001}-x}{x+1}$
$=\underset{x\to \infty }{\lim }\frac{x^{1000}-1}{1+\left(\frac{1}{x}\right)}=+\infty$