Math Problem Statement

Исследовать ряды на сходимость, применяя признак сравнения: Σ (3^n) / (4^n + 1)

Solution

The problem asks to investigate the convergence of the series:

n=13n4n+1\sum_{n=1}^{\infty} \frac{3^n}{4^n + 1}

using the comparison test ("признак сравнения").

Step 1: Simplifying the General Term

The general term of the series is:

an=3n4n+1a_n = \frac{3^n}{4^n + 1}

For large nn, the 4n4^n term in the denominator dominates the constant 1, so we can approximate:

[ a_n \approx \frac{3^n}{4^n} = \left(\frac{3}{4}\right)^n \

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Math Problem Analysis

Mathematical Concepts

Series
Convergence
Comparison Test

Formulas

a_n = 3^n / (4^n + 1)
(3/4)^n for comparison

Theorems

Comparison Test for Series Convergence

Suitable Grade Level

University Level