Determine the X values for which the power series of n=0 to infinity N!X^N converge and the power series of n=0 to infinity X^N/N!

Step-by-step explanation:

∑ₙ₌₀°° n! xⁿ

Use ratio test:

L = lim(n→∞)│aₙ₊₁ / aₙ│

L = lim(n→∞)│[(n+1)! xⁿ⁺¹] / (n! xⁿ)│

L = lim(n→∞)│[(n+1)! / n!] (xⁿ⁺¹ / xⁿ)│

L = lim(n→∞)│(n+1) x│

L = ∞

The series is divergent for all values of x.

∑ₙ₌₀°° xⁿ / n!

Use ratio test:

L = lim(n→∞)│aₙ₊₁ / aₙ│

L = lim(n→∞)│[xⁿ⁺¹ / (n+1)!] / (xⁿ / n!)│

L = lim(n→∞)│[xⁿ⁺¹ / (n+1)!] (n! / xⁿ)│

L = lim(n→∞)│(xⁿ⁺¹ / xⁿ) [n! / (n+1)!]│

L = lim(n→∞)│x / (n+1)│

L = 0

The series is convergent for all values of x.

x =-.20752

step-by-step explanation:

4^(x+2)=12

4^(x+2)=12

take the log base 4 on each side

log 4 (4^(x+2) = log 4(12)

we know log b(a^y) = y log b (a)

(x+2) log 4(4) = log 4 (12)

x+2 = log 4 (12)

we know   logb c = loga c/loga b

we want to convert to base 10   so a = 10

by default, we do not write the 10   c = 4   and b = 10

log4 10 = log 12/log 4

log4 10 = log 12/ log 4

x+2 = log 12/ log 4

subtract 2 from each side

x +2 -2 = log 12/ log 4 -2

x = 1.79248125 -2

x =-.20752