Solving A Nonstandard Equation in Two Ways

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[rah]

There are complex solutions obtained by "complexifying" the RHS: sqrt(2) * x^(xqrt(2) ) * exp( sqrt(2) * x^(xqrt(2) ) ) = sqrt(2) * exp(sqrt(2) ) * exp( 2 pi i n )
Then apply LambertW to both sides. Only main branch of W yields solutions.

[Qermaq]

Have you ever done a video on 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ... ? It's a well-known series but I'd be interested in learning your favorite methods for finding the sum.

[seanfraser]

Let f(x) = x^sqrt(2) +lnx -1. We’re looking for all roots of f.

f is only defined for x>0. Since f’(x) = sqrt(2)x^(sqrt(2)-1) + 1/x, f’(x) > 0 for all x>0. So f is strictly increasing and thus can have no more than one root.

By inspection, x=1 is a root. So x=1.

[Qermaq]

This one looks trickier than it is. Once I noticed 1 is a solution, it made me think of domains. x cannot be negative or else the equation makes no real sense. As x goes from 1 toward 0 ln(x) decreases faster than pi/4, and as it goes up it levels out. But taking a number to a power greater than 1 does kinda the opposite. It leaves the neighborhood of 0 fairly level and the slope gets steeper. So because the lines are curved opposite to one another you won't get another intersection.

[scottleung]

I just guessed and checked by plugging in x=1.

[mcwulf]

I am getting used to these! Solved it with a bit of manipulation.

[bramont]

Gracias profesor por el tiempo que nos dedica y espero que tenga una feliz navidad en compañía de sus seres queridos.