Question

Convergence Tests for Series: Fill in the blanks.

The Divergence Test: If the sequence $\left\{a_k\right\}$does not converge to ____, then the series$\sum _{k=1}^{\infty }{a}_k$__.

Short answer

If the sequence $\left\{a_k\right\}$does not converge to 0, then the series $\sum _{k=1}^{\infty }{a}_k$diverges.

Step-by-step solution

Step 1. Given Information

The given test is the divergence test.

Step 2. Explanation

• According to the divergence test, if the sequence $\underset{k\to \infty }{\lim }\left\{a_k\right\}$does not exist or exist but is not equal to 0, then the series $\sum _{k=1}^{\infty }{a}_k$is divergent.
• So, if $\left\{a_k\right\}$does not converge to 0, the series is divergent.