Q. 4 - Calculus

Question

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

 Find an example of a divergent series of the form 

(a) that satisfies conditions (i) and (iii), but not condition (ii); 

(b) that satisfies conditions (i) and (ii), but not condition (iii). 

Step-by-Step Solution

Answer

(i) $$\begin{gathered} \left(i\right)T\mathrm{he} \mathrm{series} \sum _{\mathrm{k}=1}^{\infty }{(-1)}^{\mathrm{k}+1}{\mathrm{a}}_{\mathrm{k}} \\ \mathrm{satisfying} \mathrm{the} \mathrm{given} \mathrm{condition} \mathrm{is} \sum _{\mathrm{k}=1}^{\infty }\frac{{(-1)}^{\mathrm{k}+1}}{2^{\mathrm{k}}}. \\ \left(ii\right), The \mathrm{series} \sum _{\mathrm{k}=1}^{\infty }{(-1)}^{\mathrm{k}+1}{\mathrm{a}}_{\mathrm{k}} \\ \mathrm{satisfying} \mathrm{the} \mathrm{given} \mathrm{condition} \mathrm{is} \sum _{k=1}^{\infty }\frac{{(-1)}^{k+1}k}{2k-1^{}}. \end{gathered}$$

Step 1. Given

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

Step 2. Hypothesis of alternating series

$$\begin{gathered} \mathrm{It} \mathrm{states} \mathrm{that} \\ \mathrm{if} \left\{{\mathrm{a}}_{\mathrm{k}}\right\} \mathrm{is} \mathrm{a} \mathrm{sequence} \mathrm{of} \mathrm{positive} \mathrm{number} \mathrm{with} {\mathrm{a}}_{\mathrm{k}+!}<{\mathrm{a}}_{\mathrm{k}} \mathrm{for} \mathrm{every} \mathrm{k}\ge 1 \mathrm{and} \underset{\mathrm{k}\to \infty }{\lim } {\mathrm{a}}_{\mathrm{k}}=0 \\ \mathrm{then} \mathrm{alternating} \mathrm{series} \sum _{\mathrm{k}=1}^{\infty }{(-1)}^{\mathrm{k}+1}{\mathrm{a}}_{\mathrm{k}} \mathrm{and} \sum _{\mathrm{k}=1}^{\infty }{(-1)}^{\mathrm{k}}{\mathrm{a}}_{\mathrm{k}}\mathrm{both} \mathrm{converge} \mathrm{is} \left|{\mathrm{a}}_{\mathrm{k}+1}\right|<\left|{\mathrm{a}}_{\mathrm{k}}\right| \end{gathered}$$

Part(a) Step 3. Explanation

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{series}\sum _{\mathrm{k}=1}^{\infty }{(-1)}^{k+1}{a}_k=\sum _{k=1}^{\infty }\frac{{(-1)}^{k+1}}{2^k} \\ \mathrm{The} \mathrm{sequence} \left\{{\mathrm{a}}_{\mathrm{k}}\right\}=\left\{\frac{1}{2^{\mathrm{k}}}\right\} \mathrm{is} \mathrm{a} \mathrm{positive} \mathrm{terms} \\ \mathrm{The} \mathrm{value} \mathrm{of} \underset{\mathrm{k}\to \infty }{\lim }{\mathrm{a}}_{\mathrm{k}}=\underset{\mathrm{k}\to \infty }{\lim }\left(\frac{1}{\mathrm{k}}\right) \\ =0 \\ \mathrm{But} \mathrm{the} \mathrm{series} \sum _{\mathrm{k}=1}^{\infty }{(-1)}^{\mathrm{k}+1}{\mathrm{a}}_{\mathrm{k}}=\sum _{\mathrm{k}=1}^{\infty }\frac{{(-1)}^{\mathrm{k}+1}}{2^{\mathrm{k}}} \mathrm{is} \mathrm{an} \mathrm{increasing} \mathrm{series} . \\ \mathrm{Thus}, \mathrm{the} \mathrm{series} \sum _{\mathrm{k}=1}^{\infty }{(-1)}^{\mathrm{k}+1}{\mathrm{a}}_{\mathrm{k}} \mathrm{satisfying} \mathrm{the} \mathrm{given} \mathrm{condition} \mathrm{is} \sum _{\mathrm{k}=1}^{\infty }\frac{{(-1)}^{\mathrm{k}+1}}{2^{\mathrm{k}}}. \end{gathered}$$

Part(b) Step 4. Explanation

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{series}\sum _{\mathrm{k}=1}^{\infty }{(-1)}^{k+1}{a}_k=\sum _{k=1}^{\infty }\frac{{(-1)}^{k+1}k}{2k-1^{}} \\ \mathrm{The} \mathrm{sequence} \left\{{\mathrm{a}}_{\mathrm{k}}\right\}=\left\{\frac{k}{2k-1^{}}\right\} \mathrm{is} \mathrm{a} \mathrm{positive} \mathrm{terms} \\ \mathrm{The} \mathrm{value} \mathrm{of} \underset{\mathrm{k}\to \infty }{\lim }{\mathrm{a}}_{\mathrm{k}}=\underset{\mathrm{k}\to \infty }{\lim }\left(\frac{k}{2k-1}\right) \\ =\frac{1}{2} \\ \mathrm{Thus}, \mathrm{the} \mathrm{series} \sum _{\mathrm{k}=1}^{\infty }{(-1)}^{\mathrm{k}+1}{\mathrm{a}}_{\mathrm{k}} \mathrm{satisfying} \mathrm{the} \mathrm{given} \mathrm{condition} \mathrm{is} \sum _{k=1}^{\infty }\frac{{(-1)}^{k+1}k}{2k-1^{}}. \end{gathered}$$