We analyze the problem of estimation of the error of approximation of a branched continued fraction, which is a multidimensional generalization of a continued fraction. By the method of fundamental inequalities, we establish truncation error bounds for the branched continued fraction

$$ {\sum}_{i_{1=1}}^N\frac{a_{i(1)}}{1}+{\sum}_{i_{2=1}}^{i_1}\frac{a_{i(2)}}{1}+{\sum}_{i_{3=1}}^{i_2}\frac{a_{i(3)}}{1}+\cdots $$

whose elements belong to certain rectangular sets in the complex plane. The obtained results are applied to multidimensional S- and A-fractions with independent variables.

中文翻译：

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分支连续分数的截断误差界∑ i 1 = 1 N ai 1 1 + ∑ i 2 = 1 i 1 ai 2 1 + ∑ i 3 = 1 i 2 ai 3 1 +⋯$$ {\ sum} _ {i_ {1 = 1}} ^ N \ frac {a_ {i（1）}} {1} + {\ sum} _ {i_ {2 = 1}} ^ {i_1} \ frac {a_ {i（2）} } {1} + {\ sum} _ {i_ {3 = 1}} ^ {i_2} \ frac {a_ {i（3）}} {1} + \ cdots $$

我们分析了分支连续分数的逼近误差的估计问题，这是连续分数的多维概括。通过基本不等式的方法，我们为分支的连续分数建立了截断误差界

$$ {\ sum} _ {i_ {1 = 1}} ^ N \ frac {a_ {i（1）}} {1} + {\ sum} _ {i_ {2 = 1}} ^ {i_1} \ frac {a_ {i（2）}} {1} + {\ sum} _ {i_ {3 = 1}} ^ {i_2} \ frac {a_ {i（3）}} {1} + \ cdots $$

其元素属于复平面中的某些矩形集。将获得的结果应用于具有自变量的多维S-和A-分数。