Question

Explain the term alternating series.

Short answer

$\sum _{\mathrm{k}=1}^{\infty }\frac{{(-1)}^{\mathrm{k}+1}}{\mathrm{k}}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}..........$

Step-by-step solution

Step 1.  To explain 

The term alternating series.

Step 2. Explanation

$$\begin{gathered} \mathrm{Let}{\mathrm{a}}_{\mathrm{k}}\mathrm{be}\mathrm{the}\mathrm{sequence}\mathrm{of}\mathrm{positive}\mathrm{integers}. \\ \mathrm{The}\mathrm{alternating}\mathrm{series}\mathrm{is}\mathrm{in}\mathrm{the}\mathrm{form}\sum _{\mathrm{k}=1}^{\infty }{(-1)}^{\mathrm{k}-1}{\mathrm{a}}_{\mathrm{k}}\mathrm{or}\sum _{\mathrm{k}=1}^{\infty }{(-1)}^{\mathrm{k}}{\mathrm{a}}_{\mathrm{k}} \\ \mathrm{Consider}\mathrm{the}\mathrm{series}\sum _{\mathrm{k}=1}^{\infty }\frac{{(-1)}^{\mathrm{k}+1}}{\mathrm{k}} \\ \mathrm{Expanding}\sum _{\mathrm{k}=1}^{\infty }\frac{{(-1)}^{\mathrm{k}+1}}{\mathrm{k}}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}.......... \\ \mathrm{Hence}\mathrm{the}\mathrm{series}\mathrm{is}\mathrm{alternating}\mathrm{series}. \end{gathered}$$