Question

$$\begin{gathered} \mathrm{Consider}\mathrm{the}\mathrm{series}\sum _{k=1}^{\infty }{(-1)}^{k+1}{a}_{k}and\sum _{k=1}^{\infty }{(-1)}^k{a}_k \\ \mathrm{where} \\ \left(\mathrm{i}\right)\left\{{\mathrm{a}}_{\mathrm{k}}\right\}\mathrm{is}\mathrm{a}\mathrm{sequence}\mathrm{of}\mathrm{positive}\mathrm{number} \\ \left(\mathrm{ii}\right)\mathrm{the}\mathrm{sequence}\left\{{\mathrm{a}}_{\mathrm{k}}\right\}\mathrm{is}\mathrm{strictly}\mathrm{decreasing} \\ \left(\mathrm{iii}\right)\underset{\mathrm{k}\to \infty }{\lim }{\mathrm{a}}_{\mathrm{k}}=0 \end{gathered}$$

A series can fail two of the three conditions, (i), (ii) and (iii), and still converge. Which of the three conditions must an alternating series pass in order to converge?

Short answer

According to the Divergence Test if the sequence {a_k} does not converge to zero, then the

series $\sum _{k=1}^{\infty }{a}_k$, diverges.

Hence, for the series$\sum _{k=1}^{\infty }{(-1)}^{k+1}{a}_{k}and\sum _{k=1}^{\infty }{(-1)}^k{a}_k$, to be convergent it must pass the third condition.

Step-by-step solution

Step 1. Given 

$$\begin{gathered} \mathrm{Given}\mathrm{the}\mathrm{series}\sum _{k=1}^{\infty }{(-1)}^{k+1}{a}_{k}and\sum _{k=1}^{\infty }{(-1)}^k{a}_k \\ \mathrm{where} \\ \left(\mathrm{i}\right)\left\{{\mathrm{a}}_{\mathrm{k}}\right\}\mathrm{is}\mathrm{a}\mathrm{sequence}\mathrm{of}\mathrm{positive}\mathrm{number} \\ \left(\mathrm{ii}\right)\mathrm{the}\mathrm{sequence}\left\{{\mathrm{a}}_{\mathrm{k}}\right\}\mathrm{is}\mathrm{strictly}\mathrm{decreasing} \\ \left(\mathrm{iii}\right)\underset{\mathrm{k}\to \infty }{\lim }{\mathrm{a}}_{\mathrm{k}}=0 \end{gathered}$$

Step 2. Explanation

If the third condition is not passed by an alternating series , then $\underset{k\to \infty }{\lim }{a}_k\varnothing =0$.

According to the Divergence Test if the sequence {a_k} does not converge to zero, then the

series localid="1649138812346" $\sum _{k=1}^{\infty }{a}_k$, diverges.

Hence, for the serieslocalid="1649138756493" $\sum _{k=1}^{\infty }{(-1)}^{k+1}{a}_{k}and\sum _{k=1}^{\infty }{(-1)}^k{a}_k$, to be convergent it must pass the third condition.