Determine whether the series ∞ ln(2n) 3n n=1 converges or diverges. solution the function f(x) = ln(2x) (3x) is positive and continuous for x > because the logarithm function is continuous. but it is not obvious whether or not f is decreasing, so we compute its derivative: f '(x) = 3x − 3 ln(2x) 9x2 = 3x2 .

there is only one way to arrange all the eight relative to each other because they are not distinct. we have

now there are nine places to place an (and not more than one in one place).   because we have four symbols, we need to choose four of these nine places to insert an . order does not matter because the are not distinct.

thus there are 1 · c(9,4) ways, or, if we calculate it,

thus there are 126 ways to form such "words."

use a calculator or a graph