Week 10 – Radicals, Arithmetic Series and Infinite Geometric Series

Posted on by shannonf2015

During review, I have recalled some concepts I have forgotten.

One of them is when a value or variable has an exponent that is a fraction. When it’s a fraction, the denominator represents the root. We can remember this as Flower Power.

Example:

64^\frac{1}{3} —> \sqrt[3]{64} —> 4

Another simple method I recall is adding an arithmetic series that is finite. The first step is taking the first term and the last term in the series and adding them together. Next, count how many terms there are in total and divide the amount by two. Multiply the terms divided by two with the sum of both first and last terms.

Example:

10 + 5 + 0 -5 -10 -15

  1. 10 + (-15) = -5
  2. Number of terms in total: 6
  3. \frac{6}{2} = 3
  4. (-5)(3)
  5. Sum = -15

Another concept I did not talk about last time was infinite geometric series. We can tell if they converge or diverge by looking at the value of the common ratio.

Diverges: r > 1; no sum

Converges:  -1 < r < 1

Formula:

Image result for infinite geometric series formula

Example:

2 + 3 + 4.5 + 6.75 +…

r = \frac{3}{2}

The common ratio is greater than 1, so it is infinite (diverges); no sum.

-0.5 – 0.05 – 0.005 – 0.0005 – …

r = \frac{-0.05}{-0.5} = 0.1

The common ratio is greater than -1 and smaller than 1 so it is finite (converges); has a sum.

Let’s look at an example of how to use the formula with a geometric series.

8 + 2 + 0.5 + 0.125 + …

Information given:

  • a = 8
  • r = \frac{1}{4}

Plug in the information given in the formula and solve.

Remember to flip the fraction (reciprocals) when doing division.