Q. 7 - Calculus

Question

 Explain why the sum of a series satisfying the hypotheses of the alternating series test is between any two consecutive terms in its sequence of partial sums. 

Step-by-Step Solution

Answer

The sum of a series satisfying the hypotheses of the alternating series test is between any two consecutive terms in its sequence of partial sums because signs of the terms are alternating.

The magnitudes are decreasing because the series is monotonically decreasing, the terms of the sequence of partial sums gives the property that the sum of a series satisfying the hypotheses of the alternating series test is between any two consecutive terms in its sequence of partial sums.

Step 1. Given

The given series is $\sum _{k=1}^{\infty }{(-1)}^{k+1}{a}_k$.

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

The sum of a series satisfying the hypotheses of the alternating series test is between any two consecutive terms in its sequence of partial sums because signs of the terms are alternating.

The magnitudes are decreasing because the series is monotonically decreasing, the terms of the sequence of partial sums gives the property that the sum of a series satisfying the hypotheses of the alternating series test is between any two consecutive terms in its sequence of partial sums.