Problem 43 - Essential Calculus

Question

Use the Direct Comparison Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{1}{3^{n}+1} $$

Step-by-Step Solution

Answer

The series \(\sum_{n=0}^{\infty} \frac{1}{3^{n}+1}\) converges.

Step 1: Identify a known series for comparison

Since the general term in the series is \(\frac{1}{3^n +1}\), choosing the series \(\sum_{n=0}^{\infty}\frac{1}{3^n}\) for comparison purposes is reasonable. The convergence of this series is known because it is a geometric series with a common ratio \(r = \frac{1}{3}\) where \(|r|<1\). Hence, it converges.

Step 2: Apply the Direct Comparison Test

To apply the Direct Comparison Test, we must show that the terms of our series are always less than or equal to the terms of the convergent series we've chosen for comparison. We have \(\frac{1}{3^n+1}\) and \(\frac{1}{3^n}\). Since we're adding 1 in the denominator of the former, it will always be less than or equal to the latter for all n, so we have \(\frac{1}{3^n+1} \leq \frac{1}{3^n}\). This inequality proves that the terms of our series are less compared to the corresponding terms of the converging series.

Step 3: Conclude with the result

Since the terms of our series are always less than or equal to the terms of a known converging series, by the Direct Comparison Test, the original series \(\sum_{n = 0}^{\infty}\frac{1}{3^n+1}\) must also converge.