Chapter 12, Problem 42RE

To determine

Calculate the sum of the ${\displaystyle \sum _{j=2}^{100}\frac{1}{j-1}}$ without using sigma notation and do not evaluate it.

Answer to Problem 42RE

The final summation of ${\displaystyle \sum _{j=2}^{100}\frac{1}{j-1}}$ is $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\mathrm{........}+\frac{1}{99}$ .

Explanation of Solution

Given information:

It is given in the question that, ${\displaystyle \sum _{j=2}^{100}\frac{1}{j-1}}$

By evaluating the given summation ${\displaystyle \sum _{j=2}^{100}\frac{1}{j-1}}$ ,

by using the method of expanding

  $\begin{array}{l}\frac{1}{2-1}+\frac{1}{3-1}+\frac{1}{4-1}+\frac{1}{5-1}+\frac{1}{6-1}+\mathrm{..........}+\frac{1}{100-1}\\ \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\mathrm{........}+\frac{1}{99}\end{array}$

The final summation of ${\displaystyle \sum _{j=2}^{100}\frac{1}{j-1}}$ is $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\mathrm{........}+\frac{1}{99}$ .