An easy method?

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1 Answer
Jim H
Jan 21, 2018

Hint: 1/(sqrtn+sqrt(n+1)) = "what"1√n+√n+1=what?

Explanation:

Rationalize the denominators:

1/((sqrtn+sqrt(n+1))) * ((sqrtn - sqrt(n+1)))/((sqrtn-sqrt(n+1))) = (sqrtn-sqrt(n+1))/(n-(n+1))1(√n+√n+1)⋅(√n−√n+1)(√n−√n+1)=√n−√n+1n−(n+1)

= sqrt(n+1)-sqrtn=√n+1−√n

So the sum above is equivalent to

(sqrt2 - 1) + (sqrt3 - sqrt2) +(sqrt4-sqrt3) + * * * +(sqrt8-sqrt7) + (sqrt9 - sqrt 8)(√2−1)+(√3−√2)+(√4−√3)+⋅⋅⋅+(√8−√7)+(√9−√8)
Which is -1+sqrt9 = 2−1+√9=2

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