DAILY MATH CHALLENGE THREAD #4

Oh I see, there's a much nicer solution though.

Correct!

Yeah that's the idea

[eqn]S = \sum_{n = 1}^{\infty} \frac{F_n}{5^n} = \frac{1}{5} + \frac{1}{25} + \frac{2}{125} + \frac{3}{625} + ...[/eqn]
[eqn]\frac{S}{5} = \frac{1}{25} + \frac{1}{125} + \frac{2}{625} + \frac{3}{3125} ...[/eqn]
[eqn]\frac{4S}{5} = \frac{1}{5} + \frac{1}{125} + \frac{1}{625} + \frac{2}{3125} + ...[/eqn]
[eqn]\frac{4S}{5} = \frac{1}{5} + \frac{S}{25}[/eqn]
[eqn]\frac{19S}{25} = \frac{1}{5}[/eqn]
[eqn]S = \frac{5}{19}[/eqn]

Here's my work

I guess its fairly obvious the series converges. For example, by induction we have that F_n is at most say 2^n. So this method of "solving for S" is valid since S is a finite real number

b) Evaluate [eqn]\sum_{n=1}^\infty \frac{F_n}{N^n}[/eqn]

Is that supposed to be a lowercase "n" on the bottom? If so I doubt this one has any kind of nice answer

define the predecessor function in the untyped lambda calculus

Is there even a closed form for [eqn]\sum_{n=1} ^{\infty} \frac{1}{n^{n}}[/eqn]

sort of
en.wikipedia.org/wiki/Sophomore's_dream

Here's a kinda stupid one, but hopefully one of you finds it amusing. Let X be a non-empty set. Show that X admits a poset structure with a greatest element without using AC

for you, my children