Question 1 and 2, Exercise 4.7

Solutions of Question 1 and 2 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Evaluate the sum: $\sum_{k=1}^{5} \frac{1}{2 k}$

Solution.

$$\begin{align*} \sum_{k=1}^{5} \frac{1}{2k} &= \frac{1}{2(1)} + \frac{1}{2(2)} + \frac{1}{2(3)} + \frac{1}{2(4)} + \frac{1}{2(5)}\\ &= \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10}\\ &= \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10}\\ &= \frac{30}{60} + \frac{15}{60} + \frac{10}{60} + \frac{7.5}{60} + \frac{6}{60}\\ &= \frac{68.5}{60} \approx 1.14\end{align*}$$

Evaluate the sum: $\sum_{k=1}^{6} \frac{1}{2 k+1}$

Solution.

$$\begin{align*} \sum_{k=1}^{6} \frac{1}{2k+1} &= \frac{1}{2(1)+1} + \frac{1}{2(2)+1} + \quad\frac{1}{2(3)+1} \\ &+ \frac{1}{2(4)+1} + \frac{1}{2(5)+1} + \frac{1}{2(6)+1}\\ &= \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} + \frac{1}{13}\\ &= \frac{35}{105} + \frac{21}{105} + \frac{15}{105} + \frac{11.67}{105} + \frac{9.55}{105} + \frac{8.08}{105}\\ &= \frac{100.3}{105} \approx 0.96 \end{align*}$$

Question 3 & 4 >