Question

Sigma Notation Write the sum without using sigma notation.

$\sum _{j=1}^4\sqrt{\frac{j-1}{j+1}}$

Short answer

The sum is$\sqrt{\frac{0}{2}}+\sqrt{\frac{1}{3}}+\sqrt{\frac{2}{4}}+\sqrt{\frac{4}{5}}.$

Step-by-step solution

Step 1.  Given

The given sequence is$\sum _{j=1}^4\sqrt{\frac{j-1}{j+1}}.$

Step 2.  To determine

We have to write the sum without using sigma notation.

Step 3.  Calculation

We’ll plug the values for j, to get rid of the sigma notation.

$\begin{array}{l}\sum _{j=1}^4\sqrt{\frac{j-1}{j+1}}\\ =\sqrt{\frac{1-1}{1+1}}+\sqrt{\frac{2-1}{2+1}}+\sqrt{\frac{3-1}{3+1}}+\sqrt{\frac{4-1}{4+1}}\\ =\sqrt{\frac{0}{2}}+\sqrt{\frac{1}{3}}+\sqrt{\frac{2}{4}}+\sqrt{\frac{4}{5}}\end{array}$

Hence the sum is$\sqrt{\frac{0}{2}}+\sqrt{\frac{1}{3}}+\sqrt{\frac{2}{4}}+\sqrt{\frac{4}{5}}.$