Chapter 12, Problem 5CT

To determine

To find: The sequence.

Answer to Problem 5CT

The sequence is $-\frac{2}{5}+\frac{3}{6}-\frac{4}{7}+\mathrm{......}+\frac{11}{14}={\displaystyle \sum _{k=1}^{10}{\left(-1\right)}^k}\left(\frac{k+1}{k+4}\right)$ .

Explanation of Solution

Given information:

The given sequence is $-\frac{2}{5}+\frac{3}{6}-\frac{4}{7}+\mathrm{......}+\frac{11}{14}$ .

Calculation:

The given equation $-\frac{2}{5}+\frac{3}{6}-\frac{4}{7}+\mathrm{......}+\frac{11}{14}$ is having the form ${\left(-1\right)}^2\left(\frac{k+1}{k+4}\right)$ and the odd terms are negative and the even terms are positive.

The sum starts at $s_1=-\frac{2}{5}$ and ends at $s_k=\frac{11}{14}$ .

To find the number of terms equate ${\left(-1\right)}^2\left(\frac{k+1}{k+4}\right)$ to $\frac{11}{14}$ .

Thus, the number of terms is $10$ .

Therefore, the sequence is $-\frac{2}{5}+\frac{3}{6}-\frac{4}{7}+\mathrm{......}+\frac{11}{14}={\displaystyle \sum _{k=1}^{10}{\left(-1\right)}^k}\left(\frac{k+1}{k+4}\right)$ .