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The problem asks to investigate the convergence of the series:

\[
\sum_{n=1}^{\infty} \frac{3^n}{4^n + 1}
\]

using the comparison test ("признак сравнения").

### Step 1: Simplifying the General Term

The general term of the series is:

\[
a_n = \frac{3^n}{4^n + 1}
\]

For large \( n \), the \( 4^n \) 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
\

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

Investigate the convergence of the series Σ (3^n) / (4^n + 1) using the comparison test. Learn how to simplify the general term and approximate it for large n. This is a common problem in university-level calculus, focusing on series and convergence tests.

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