Question

Determine whether the following statement is true or false. If the statement is true, explain why. If the statement is false, provide a counterexample. Make sure that counterexample is "obvious".

If converges, then the series converges.

Answer 1

The given statement is false.

Let's clarify why this is the case with an example.

For our counterexample, let's consider the harmonic series, which is defined as: 1 + 1/2 + 1/3 + 1/4 + ... . This is a sequence where the terms progressively converge to zero as the number of terms goes to infinity.

Yet, despite the individual terms getting smaller and smaller, the sum of the terms (which is what constitutes the series) does not converge. Irrespective of how many terms you add together, the series never reaches a fixed, final amount. Instead, it continues going to infinity.

So even if a sequence converges, it doesn't guarantee that the series also converges. Therefore, the given statement, "If a sequence converges, then the series also converges", is proven false.