Your assignment is to shade a disk of radius 1. Then your friend paints a white...

Transcribed Image Text:

Your assignment is to shade a disk of radius 1. Then your friend paints a white disk of radius 1/2 over it. You shade a disk of radius 1/3 on top of that, and your friend paints a white disk of radius 1/4 on top of that... As you continue, the shade portion looks more and more like a target. Figure: Painting a target by shading concentric disks blue then white ad infinitum If you could continue this forever (with really small paintbrushes!), how much of the circle would be shaded in the limit? Your answer must include the following:| 1. Calculate the shaded area in each step of the process 2. Find an expression for the sum of the first n terms, S. 3. Prove that the limit of the partial sums exist and therefore the series converges. 4. How many terms are needed to approximate the alternating series to within 0.0001 of its actual value? Your assignment is to shade a disk of radius 1. Then your friend paints a white disk of radius 1/2 over it. You shade a disk of radius 1/3 on top of that, and your friend paints a white disk of radius 1/4 on top of that... As you continue, the shade portion looks more and more like a target. Figure: Painting a target by shading concentric disks blue then white ad infinitum If you could continue this forever (with really small paintbrushes!), how much of the circle would be shaded in the limit? Your answer must include the following:| 1. Calculate the shaded area in each step of the process 2. Find an expression for the sum of the first n terms, S. 3. Prove that the limit of the partial sums exist and therefore the series converges. 4. How many terms are needed to approximate the alternating series to within 0.0001 of its actual value?