Q. 6 - Calculus

Question

 Explain why a series satisfying the hypotheses of the alternating series test has a sum with the same sign as the first term in the series.

Step-by-Step Solution

Answer

This implies that the magnitude of the successive terms decreases, which implies that the magnitude of the first term of the series is largest.

Therefore, the sum of the series will have the same sign as the first term. Hence, proved.

Step 1. Given

$\sum _{k=1}^{\infty }{(-1)}^{k+1}{a}_k and \sum _{k=1}^{\infty }{(-1)}^k{a}_k$

Step 2. Hypothesis of alternating series

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

Step 3. Explanation

This implies that the magnitude of the successive terms decreases, which implies that the magnitude of the first term of the series is largest.

Therefore, the sum of the series will have the same sign as the first term. Hence, proved.