Have no idea how to solve this!!?

Question

1/(√1 + √3) + 1/(√3+√5) + 1/(√5+√7) +…+ 1/[√(2n-1) +√(2n+1)] = 100

. Solve for n
. Find n for the same sum = L (instead of 100)
I'm in the Log Chapter. So I think think prob should be solve by using log.
Thanks for your help

Answer 1

Rationalize all of the denominators. For example, for the first term, multiply its numerator and denominator by (√3 - √1) and simplify the denominator.

Just look at, say, the first three terms, rationalizing them as described above, and you'll see what's going on. This is sometimes called a "telescoping sum". No logs needed.

Answer 2

The nth term can be written as
[√(2n+1) - √(2n-1)]/(2)
It can be seen that √(2n+1) > √(2n-1) so numerator is always positive and each term is positive and less than the preceding term.

Thus the series converges as n --> infinity with a limit of 0.

Thus the sum can never be 100