Q. 5 - Calculus

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? 

Step-by-Step Solution

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