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

Posted about 2 months ago

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.

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