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Q: 设 $|x|<1$, 求 $\lim _{x \rightarrow \infty}(1+x)\left(1+x^{2}\right)\left(1+x^{3}\right)\left(1+x^{4}\right) \cdots\left(1+x^{n}\right)$

∻Solution by Steps∻ ‖ step 1 ⋮ We start with the limit expression: $ \lim_{x \rightarrow \infty} (1+x)(1+x^2)(1+x^3)(1+x^4) \cdots (1+x^n) $ ‖ ‖ step 2 ⋮ As $x$ approaches infinity, each term $(1+x^k)$ behaves like $x^k$ for large $x$. Thus, we can approximate the product: $ \lim_{x \rightarrow \infty} x^{1+2+3+\cdots+n} = x^{\frac{n(n+1)}{2}} $ ‖ ‖ step 3 ⋮ Therefore, the limit simplifies to: $ \lim_{x \rightarrow \infty} x^{\frac{n(n+1)}{2}} = \infty $ for $n \geq 1 $ ‖ ‖ step 4 ⋮ Hence, the final result is: $ \lim_{x \rightarrow \infty} (1+x)(1+x^2)(1+x^3)(1+x^4) \cdots (1+x^n) = \infty $ ‖ ∻Answer∻ ⚹ The limit diverges to infinity. ⚹ ∻Key Concept∻ ⚹ Understanding limits involving products of polynomials as variables approach infinity. ⚹ ∻Explanation∻ ⚹ As $x$ increases, the dominant terms in the product dictate the behavior of the limit, leading to divergence. ⚹◊What is the behavior of the limit as $ n $ approaches infinity for the expression $ (1+x)(1+x^2)(1+x^3)\cdots(1+x^n) $ given \( |x|

Question

设 $|x|<1$, 求 $\lim _{x \rightarrow \infty}(1+x)\left(1+x^{2}\right)\left(1+x^{3}\right)\left(1+x^{4}\right) \cdots\left(1+x^{n}\right)$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We start with the limit expression: $ \lim_{x \rightarrow \infty} (1+x)(1+x^2)(1+x^3)(1+x^4) \cdots (1+x^n) $ ‖ ‖ step 2 ⋮ As $x$ approaches infinity, each term $(1+x^k)$ behaves like $x^k$ for large $x$. Thus, we can approximate the product: $ \lim_{x \rightarrow \infty} x^{1+2+3+\cdots+n} = x^{\frac{n(n+1)}{2}} $ ‖ ‖ step 3 ⋮ Therefore, the limit simplifies to: $ \lim_{x \rightarrow \infty} x^{\frac{n(n+1)}{2}} = \infty $ for $n \geq 1 $ ‖ ‖ step 4 ⋮ Hence, the final result is: $ \lim_{x \rightarrow \infty} (1+x)(1+x^2)(1+x^3)(1+x^4) \cdots (1+x^n) = \infty $ ‖ ∻Answer∻ ⚹ The limit diverges to infinity. ⚹ ∻Key Concept∻ ⚹ Understanding limits involving products of polynomials as variables approach infinity. ⚹ ∻Explanation∻ ⚹ As $x$ increases, the dominant terms in the product dictate the behavior of the limit, leading to divergence. ⚹◊What is the behavior of the limit as $ n $ approaches infinity for the expression $ (1+x)(1+x^2)(1+x^3)\cdots(1+x^n) $ given $ |x| < 1⍭ Generate me a similar question◊$

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