Q. 3 - Calculus

Question

Explain the term alternating series.

Step-by-Step Solution

Answer

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

Step 1. To explain

The term alternating series.

Step 2. Explanation

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