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Q) How to find the value of:

$$\frac{1}{1-\sqrt{2}}+\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}+\dots+\frac{1}{\sqrt{101}-\sqrt{102}}$$

Ans) I am trying to find the value of this above series by rationalizing the numerator and denominator of each terms. Let $$S= \frac{1}{1-\sqrt{2}}+\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}+\dots+\frac{1}{\sqrt{101}-\sqrt{102}}$$ Therefore after rationalizing we can write this above summation as:

$$S = \left(\frac{1+\sqrt{2}}{-1}\right)+\left(\frac{\sqrt{2}+\sqrt{3}}{-1}\right)+\left(\frac{\sqrt{3}+\sqrt{4}}{-1}\right)+\dots+\left(\frac{\sqrt{101}+\sqrt{102}}{-1}\right)$$

Therefore the above summation is equal to

$$S = -\left(1+\sqrt{2}\right)-\left(\sqrt{2}+\sqrt{3}\right)-\left(\sqrt{3}+\sqrt{4}\right)-\dots-\left(\sqrt{101}+\sqrt{102}\right)$$

Now after this step I can't proceed further because none of the terms are cancelling out. Please help me out with this summation.

For e.g., we can easily find the value of

$$S = \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\dots+\frac{1}{\sqrt{101}+\sqrt{102}}$$

by rationalizing the numerator and denominator of each terms. After rationalization the series will become

$$S = \left(\sqrt{2}-1\right)+\left(\sqrt{3}-\sqrt{2}\right)+\left(\sqrt{4}-\sqrt{3}\right)+\dots+\left(\sqrt{102}-\sqrt{101}\right)$$

Now finally after cancellation the value of $S$ will be $\sqrt{102}-1$.

Egor Ivanov
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DEB SANKAR ROY
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    I think there is not much else you can do to simplify your expression. See https://math.stackexchange.com/questions/1241864/sum-of-square-roots-formula. – Alma Arjuna Jan 16 '24 at 17:36
  • Yes, I also think so. But still I gave this question in this community because I am not sure whether any finite value will exist for this above summation or not. – DEB SANKAR ROY Jan 16 '24 at 17:43
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    Its a finite summation. You can be sure its value is finite. – Alma Arjuna Jan 16 '24 at 18:03

1 Answers1

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Although the terms do not cancel, they sum up rather nicely. Note that each distinct integer root from $2$ to $101$ occurs twice. Thus, the sum is $$-1 - (2)*\sqrt{2} - 2\sqrt{3} - \cdots - 2\sqrt{101} - \sqrt{102}$$ $$=-1 - \sqrt{102}- 2\sum_{i=2}^{101} \sqrt{i}.$$

Doug
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    I think the first answer (sorted by score) in the link you mentioned considers only integer roots, so the formula doesn't really apply here. – Alma Arjuna Jan 16 '24 at 18:08