When will an infinite geometric series with –1 < r < 0 converge to a number less than the initial term? explain your reasoning, and give an example to support your answer.

If r is negative, the denominator of the formula for the sum of the series is positive and greater than 1.

If the initial term is divided by a positive number greater than 1, the result is a number smaller than the initial term.

So, if the initial term is positive, then the series will converge to a number less than the initial term.

For –1 < r < 0, an example with a1 > 0, such as 1,000 – 100 + 10 – 1 + …

a box 3 3/4 feet long or 4+ feet.

step-by-step explanation:

x= -3 and

step-by-step explanation:

to find the roots of a cubic, use division or factoring to find the factors then set equal to 0. as this is a difference of cubes, it factors into

where a is the cube root of .

here   this is 27 and the cube root of 27 is 3. so we write using the form:

.

set each factor to 0 and solve for x.

x-3=0

x=3

the quadratic expression does not factor and must be solved using the quadratic formula.

where a = 1, b=3, and c=9

this cubic has two complex roots.