Determine Series Convergence Sum of 1 over 4 plus 3 to the n

Determine the convergence or the divergence of the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{4 + 3^n}$.

In mathematical analysis, one of the key areas of study is determining the behavior of infinite series, particularly regarding their convergence or divergence. The series presented here, characterized by a general term of 1 divided by the sum of 4 and the power of 3 raised to n, poses an interesting question about this behavior. At its core, the challenge is to dissect the growth of the terms involved and ascertain whether the sum of these terms approaches a finite limit or tends towards infinity.

One effective starting point in evaluating series like this is to consider the nature of the terms as n tends to infinity. Specifically, the term 3 raised to the power of n grows exponentially, dominating the addition of the constant 4, which means the denominator tends to increase rapidly, causing the overall term to decrease. From this perspective, recognizing how such exponential growth influences series convergence can highlight the practical application of certain convergence tests.

In this case, analytical tools such as the comparison test or ratio test may come into play, each providing a systematic approach to determine convergence. By comparing the given series to a known benchmark series, we can gain insights into its behavior. This problem provides an excellent opportunity to apply these abstract tests, exploring how growth rates between the given series and a simpler series can be compared to draw conclusions regarding convergence or divergence.