Question

The Alternating Series Test: Let $\left\{a_k\right\}$be a sequence of positive numbers. If role="math" localid="1649396442317" $a_{k+1}\_\_\_a_k$for every $k\ge 1$and $\underset{k\to \infty }{\lim }{a}_k=\_\_\_$, then role="math" localid="1649396524843" $\sum _{k=1}^{\infty }{(-1)}^{k+1}{a}_k=\_\_\_$.

Short answer

Let $\left\{a_k\right\}$be a sequence of positive numbers. If $a_{k+1}\le a_k$for every $k\ge 1$and $\underset{k\to \infty }{\lim }{a}_k=0$, then $\sum _{k=1}^{\infty }{(-1)}^{k+1}{a}_k$ converges .

Step-by-step solution

Step 1. Given Information

The given test is the alternating series test.

Step 2. Explanation

• The alternating series test is performed on the monotonically decreasing sequences.
• The limit of the sequence must be 0.
• Then the series defined by$\sum _{k=1}^{\infty }{(-1)}^{k+1}{a}_k$is convergent.