Infinite Series

Each of the problems below is a link. If you click on the problem, it will take you to a YouTube video of a full, step-by-step solution for the problem. You may try these problems as practice and use the videos to check your answer, or you can just watch the videos to see more worked-out example problems.

Find the sum of the infinite geometric series 2 fifths + 4 twenty fifths + 8 one hundred twenty fifths + dot dot dot
Find the sum of the infinite geometric series 1 minus 1 fourth + 1 sixteenth minus 1 sixty fourth + dot dot dot
Find the sum of the infinite geometric series 3 halves + 9 fourths + 27 eighths + dot dot dot
A certain ball rebounds to half the height from which it is dropped. Approximate the total distance the ball travels after being dropped from 1m above the ground until it comes to rest.
A yellow square with side length 1 is divided into 9 smaller squares, and the middle one is colored blue. Then each smaller yellow square is divided into 9 smaller squares, and the middles are colored blue. If this process is continued indefinitely, what is the area that is colored blue?