SOLUTION: Find the sum of the following series: <img src="https://i.ibb.co/b6bHYcd/Code-Cogs-Eqn.gif" height = "35px">

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Question 1205331: Find the sum of the following series:


Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.
Find the sum of the following series:

~~~~~~~~~~~~~~~~~~~~~ 

Make a standard operation of "rationalizing the denominators" over each separate term/addend


    =  =  = 


    =  =  = 


    =  =  = 


    =  =  = 


    . . .    and so on  . . . 


    =  =  = 


    +-------------------------------------------------+
    |   Now add all these identities, line by line.   |
    +-------------------------------------------------+


In the right side, all the interior terms with opposite signs will cancel each other.

So, in the right side, you will get the difference   - .

In the left side, you will get the sought sum


     


    =  -  = 6 - 1 = 5.


ANSWER.  The given sum is 5.

Solved.

--------------------

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    - HOW TO rationalize a fraction by making its denominator free of square roots
    - HOW TO rationalize a fraction by making its denominator free of cubic roots
    - Amazing calculations with fractions that contain quadratic irrationalities in denominators
in this site.

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Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Each term is of the form where k ranges from k = 1 to k = 35.

Rationalize the denominator, by multiplying top and bottom by , and you should get after the simplification is done.
Therefore,
I'll leave the scratch work for the student to do.

If k = 1, then,
If k = 2, then,
If k = 3, then,
Adding those terms gets us


The inner terms and cancel out.
This will extend to the summation of terms k = 1 through k = 35
The inner terms , , ... , , cancel leaving behind

For more info and examples, search out "telescoping series".
The name is due to the fact the long series collapses to a very small expression since the inner terms cancel out.
https://mathworld.wolfram.com/TelescopingSum.html

Answer: 5
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