Question

The limit of $ \sum\limits^{1000}_{ n-1} (-1)^n \, x^n $ as $x→∞$

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

Answer 1

Answer: (C)

$\displaystyle\lim _{x \rightarrow \infty} \sum_{n=1}^{1000}(-1)^{n} x^{n}$
$=\displaystyle\lim _{x \rightarrow \infty}\left\{-x+x^{2}-x^{3}+x^{4}+\ldots+x^{1000}\right\}$
$=\displaystyle\lim _{x \rightarrow \infty}(-x) \cdot\left\{\frac{(-x)^{1000}-1}{(-x-1)}\right\}=\displaystyle\lim _{x \rightarrow \infty} \frac{x^{1001}-x}{x+1}$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{x^{1000}-1}{1+\left(\frac{1}{x}\right)}=+\infty$