Fill in the Blank Help: Solving Complex Equations with Unknown Numbers

[darko]

These two problems are a big pain (never worked a problem in this format before). # represents a number that needs to be plugged into the problem to equal the given number. All #'s are the same when you find them so in problem 1 the 4 unknown #s are not different. #=#

1.
# + 4*#^2+9*#^3+16*#^4... =6

i re wrote it so it looks like this # +2^2 * #^2 + 3^2 * #^3 = 4^2 * #^4
Looks like a taylor series. Series of 1 over 1-x? but how do i apply that?



2.
lim (1+#x)^#/x = 4
x-0

for 2 i was really confused. I was thinking of appyling the natrual log so i would get ln #/x (1+#x) and apply L'Hopital's rule to that but i don't think that's the correct method.

Sorry I am just really confused because I never worked problems like these (fill in the blank) before. Any help would be apperciated.

 

[Tide]

HINT: If

$$f(x) = \frac {1}{1-x}$$

then

$$\frac {d}{dx} \left( x \frac {df}{dx} \right) = 1 + 2x + 9x^2 + \cdot \cdot \cdot$$