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The limit of $\sum_{n=1}^{1000}(-1)^{n} x^{n}$ as $x \rightarrow \infty$
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exists and approaches to $+\infty$
$\lim _{x \rightarrow-} \sum_{n=1}^{1000}(-1)^{n} x^{n}$
$=\lim _{x \rightarrow-}\left\{-x+x^{2}-x^{3}+x^{4}+\ldots+x^{1000}\right\}$
$=\lim _{x \rightarrow \infty}(-x) \cdot\left\{\frac{(-x)^{1000}-1}{(-x-1)}\right\}=\lim _{x \rightarrow-\infty} \frac{x^{1001}-x}{x+1}$
$=\lim _{x \rightarrow-} \frac{x^{1000}-1}{1+\left(\frac{1}{x}\right)}=+\infty$
$=\lim _{x \rightarrow-}\left\{-x+x^{2}-x^{3}+x^{4}+\ldots+x^{1000}\right\}$
$=\lim _{x \rightarrow \infty}(-x) \cdot\left\{\frac{(-x)^{1000}-1}{(-x-1)}\right\}=\lim _{x \rightarrow-\infty} \frac{x^{1001}-x}{x+1}$
$=\lim _{x \rightarrow-} \frac{x^{1000}-1}{1+\left(\frac{1}{x}\right)}=+\infty$
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