Problem 89 - Essential Calculus

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(a_{n}+b_{n} \leq c_{n}\) and \(\sum_{n=1}^{\infty} c_{n}\) converges, then the series \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) both converge. (Assume that the terms of all three series are positive.)

Step-by-Step Solution

Answer

The statement is false. A counterexample can be constructed in such a way that \(a_{n}+b_{n} \leq c_{n}\) and \(\sum_{n=1}^{\infty}c_{n}\) converges, but either \(\sum_{n=1}^{\infty}a_{n}\) or \(\sum_{n=1}^{\infty}b_{n}\) or both are divergent.

Step 1: Assess the given statement

The series \(\sum_{n=1}^{\infty}c_{n}\) is given to be convergent. According to the comparison test, if \(\sum_{n=1}^{\infty}a_{n} \leq \sum_{n=1}^{\infty}c_{n}\) and \(\sum_{n=1}^{\infty}c_{n}\) is convergent, then \(\sum_{n=1}^{\infty}a_{n}\) is also convergent. The same logic applies to \(\sum_{n=1}^{\infty}b_{n}\). However, we cannot directly state that such comparisons stand because \(a_{n} + b_{n} \leq c_{n}\) does not assure that each series \(\sum_{n=1}^{\infty}a_{n}\) and \(\sum_{n=1}^{\infty}b_{n}\) individually is less than or equal to \(\sum_{n=1}^{\infty}c_{n}\)

Step 2: Provide a counterexample to prove the statement false

To show the statement is false we can provide a counterexample, choosing for instance \(a_{n} = \frac{1}{n}\), \(b_{n} = 1\), and \(c_{n} = 2\), for \(n \geq 1\). It's clear that \(a_{n}+b_{n} \leq c_{n}\) holds, and the series \(\sum_{n=1}^{\infty}c_{n}\) is convergent (being a geometric series with ratio less than 1). However, the series \(\sum_{n=1}^{\infty}b_{n} = \sum_{n=1}^{\infty}1\) is not convergent, which contradicts the original statement, showing it indeed is false.