So multiplying infinite terms to take the limit why is it possible to split?

rendongxing9 注册会员
2023-02-25 13:27

Part of this answer references GPT, GPT_Pro better solution
The limit of multiplying infinite terms can be broken because, according to the infinite multiplication theorem, as n goes to infinity, So lim(anbn) = lim(an)lim(bn), as long as an and bn converge to a certain value, you can split the original multiplier into two separate expressions, find the limit of each, and then multiply them together to find the limit of the original multiplier expression. The key here is to show that an and bn converge to a certain value, and then you can take them apart with four operations.
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cjc3751 注册会员
2023-02-25 13:27

the reference answer GPT ᴼ ᴾ ᴱ ᴺ ᴬ ᴵ < br / > in mathematics, the summation of infinity multiplication and summation of the sum of the infinite is has essential difference.

For the addition of infinite terms, generally we have no easy way to break it apart, because each term will have an effect on the result, and the addition of infinite terms does not satisfy the fundamental operation laws such as commutative and associative law, so it cannot be broken apart directly.

But the case is different for multiplying an infinite number of terms. In some cases, we can divide the multiplication of an infinite number of terms into a simpler form, and this splitting process is reasonable. For example, if we have a form like:

(1 + 1/2) ×(1 + 1/3) ×(1 + 1/4) ×... Times(1 + 1/n) times...

We can split it into two parts:

[(1 + 1/2) ×(1 + 1/4) ×(1 + 1/6) ×...] × [(1 + 1/3) ×(1 + 1/5) ×(1 + 1/7) ×...]

We find that both of the above two parentheses multiply infinite terms, but the number of terms inside each expression is finite, because the denominator inside the parentheses increases as the number of terms increases, so each term approaches zero, and finally the result of multiplying infinite terms is the result of multiplying these two expressions.

In this case, we split the original expression into two expressions, and by using the associative law of multiplying infinite terms, we make the operation feasible by turning the infinite number into a finite number. But it's important to note that not all infinite multiplications can be split this way, and in some cases, if they can be split, it requires some skill.

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2023-02-25 13:27
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2023-02-25 13:27