The limit of sum_{n=1}^{1000} (-1)^{n} x^{n} | vedclass

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(C) Let $S = \sum_{n=1}^{1000} (-1)^{n} x^{n} = -x + x^{2} - x^{3} + x^{4} - \dots + x^{1000}$.This is a finite geometric series with first term $a =

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exists and approaches to $+\infty$

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(C) Let $S = \sum_{n=1}^{1000} (-1)^{n} x^{n} = -x + x^{2} - x^{3} + x^{4} - \dots + x^{1000}$.This is a finite geometric series with first term $a = -x$, common ratio $r = -x$, and number of terms $n = 1000$.The sum is given by $S = a \frac{r^{n} - 1}{r - 1} = (-x) \frac{(-x)^{1000} - 1}{-x - 1} = (-x) \frac{x^{1000} - 1}{-(x + 1)} = x \frac{x^{1000} - 1}{x + 1} = \frac{x^{1001} - x}{x + 1}$.Now, we evaluate the limit as $x \rightarrow \infty$:$\lim_{x \rightarrow \infty} \frac{x^{1001} - x}{x + 1} = \lim_{x \rightarrow \infty} \frac{x^{1001}(1 - \frac{1}{x^{1000}})}{x(1 + \frac{1}{x})} = \lim_{x \rightarrow \infty} x^{1000} = +\infty$.

The limit of $\sum_{n=1}^{1000} (-1)^{n} x^{n}$ as $x \rightarrow \infty$ is:

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