WBJEE · Maths · Limits
The limit of \(\sum_{n=1}^{1000}(-1)^{n} x^{n}\) as \(x \rightarrow \infty\)
- A does not exist
- B exists and equals to 0
- C exists and approaches to \(+\infty\)
- D exists and approaches \(-\infty\)
Answer & Solution
Correct Answer
(C) exists and approaches to \(+\infty\)
Step-by-step Solution
Detailed explanation
\(\lim _{x \rightarrow-} \sum_{n=1}^{1000}(-1)^{n} x^{n}\) \(=\lim _{x \rightarrow-}\left\{-x+x^{2}-x^{3}+x^{4}+\ldots+x^{1000}\right\}\) \(=\lim _{x \rightarrow \infty}(-x) \cdot\left\{\frac{(-x)^{1000}-1}{(-x-1)}\right\}=\lim _{x \rightarrow-\infty} \frac{x^{1001}-x}{x+1}\)…
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