What is the solution to this infinite series problem?

[camilus]

can anyone find a solution without using a calculator??

This is the problem:


Find the positive interger k for which $$\sum \limits_{n=4}^k {1 \over \sqrt{n} + \sqrt{n+1}} = 10$$

 

[Count Iblis]

Hint: Multiply numerator and denominator by sqrt(n) - sqrt(n+1)

 

[leon1127]

I suspect this will become something call telescoping series

 

[sennyk]

Maxima

Has anyone used Maxima as a CAS? I used Maxima to check my answer to this problem and it just gives me a huge sum of radicals. When I used its float command it gave me 9.9999999999996 as the sum for my answer. Are all of the CAS's this limited?

 

[sennyk]

Rationalize for the computer

Has anyone used Maxima as a CAS? I used Maxima to check my answer to this problem and it just gives me a huge sum of radicals. When I used its float command it gave me 9.9999999999996 as the sum for my answer. Are all of the CAS's this limited?

If I rationalize the denominator first, it then gives me the correct answer. If anyone has any experiences with other CAS's, please share.

 

[robert Ihnot]

This is a very easy problem because it reduces to:

$$\sum \limits_{n=4}^k - \sqrt{n} + \sqrt{n+1}} = 10$$