How to solve an infinite radical integral using the Taylor expansion trick.

Added: 2026-05-17

[00:00:00]This integral of an infinite nested radical looks like a total nightmare.

[00:00:05]Most people would skip this instantly, but the solution is actually incredibly elegant.

[00:00:11]First, we simplify.

[00:00:13]Convert those roots into fractional exponents.

[00:00:17]Notice a pattern in the denominators?

[00:00:20]They are all factorials.

[00:00:23]This is actually the Taylor series for e raised to the power of z, where z equals 1, but we are missing the first two terms. That means our entire integrand collapses down to x raised to the power of e minus 2.

[00:00:38]Now the calculus is trivial. Just use the basic power rule. Add one to the exponent and divide by the result. We get x to the e minus 1 over e minus 1 plus our constant c. Nested infinity solved in under a minute.