Problem 84 - Essential Calculus

Question

A friend in your calculus class tells you that the following series converges because the terms are very small and approach 0 rapidly. Is your friend correct? Explain. \(\frac{1}{10,000}+\frac{1}{10,001}+\frac{1}{10,002}+\cdots\)

Step-by-Step Solution

Answer

No, the friend is not correct. The given series diverges, even though the terms are very small and approach 0 rapidly. The size of individual terms is not the determining factor for convergence of a series.

Step 1: Identify the series type

Identify the type of series given, which is a harmonic series of the form \(\sum (1/n) \)

Step 2: Apply the p-series test

Apply the p-series test. In this case, the series is of the form \( \sum (1/n) \) which is the p-series where \( p = 1 \)

Step 3: Determine convergence or divergence

Based on the p-series test, if \( p \leq 1 \), as in our case, the series diverges.

Step 4: Explain the result

Even though the terms of the sequence are getting very small very quickly, this does not guarantee that the series will converge. Since the p-value equals 1 (which is less than or equal to 1), the series diverges. Therefore, the student's friend is mistaken in asserting that the series converges solely based on the smallness of the terms.