Question

Which convergence tests can be used only on series in which the terms eventually all have the same sign?

Short answer

The alternating series test is one of the convergence tests that can be used only on series in which the terms eventually all have the same sign.

Step-by-step solution

Step 1. Given Information

Convergence tests, used only on series in which the terms eventually all have the same sign.

Step 2. Explanation

Here, we have to find that the convergence tests can be used only on series in which the terms eventually all have the same sign.

Such a test is the Alternating Series Test, which can be given as,

$$\sum_{k=0}^{\alpha }(-1)^{k}a_{k}=a_{0}-a_{1}+a_{2}-a_{3}+a_{4}-...$$, where all $$a_{k}$$ are positive or all $$a_{k}$$ are negative.

$$\implies$$ Under the conditions of alternating series test, all have the same sign.