Problem 78 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \frac{1}{200}+\frac{1}{210}+\frac{1}{220}+\frac{1}{230}+\cdots $$

Step-by-Step Solution

Answer

According to the p-series test for convergence/divergence, the given infinite series is a divergent series.

Step 1: Identify the sequence

The given series is \[ \frac{1}{200}+\frac{1}{210}+\frac{1}{220}+\frac{1}{230}+\cdots \] It can be rewritten in the general form as \[ \sum_{n=1}^{\infty} \frac{1}{{200+10n}} \]

Step 2: Recognize the series type

This series is a form of p-series. The general form of a p-series is \( \sum_{n=1}^{\infty} \frac{1}{{n^p}} \) where p is a constant. Here, we are dealing with a harmonic series since our denominator, apart from the constant, is effectively 'n', resembling the relation \( \sum_{n=1}^{\infty}\frac{1}{n} \) which is a divergence series.

Step 3: Apply the p-series test

In a p-series, the series converges when \( p>1 \) and diverges when \( p\leq1 \) . Since our denominator resembles n, which means we are essentially dealing with 1/n where p=1, this series is a divergent series, according to the p-series test.