Geometric Power Series
- Dr. Doug Davis, 3D
A Geometric Series is obtained by adding powers of a number, , all the way to infinity. For values of the limit of the sum approaching an infinite number of terms is . For other values the sum is undefined. This Geometric Power Series is obtained by adding powers of . For values of the limit of the sum obtained is . For other values the sum is undefined. This applet illustrates the partial sums where the number of terms, , can be adjusted with the slider. The terms are which can be shown by checking the "Terms" checkbox. The sum of the terms can be shown by checking the "Sum of Terms" checkbox. The "Infinite Sum" checkbox will show the infinite sum limit function .
By increasing the number of terms it can be observed that the "Sum of Terms" curve gets closer to the "Infinite Sum" curve for some values of . To help see this the "Error at Point" check box will provide a point on the -axis that can be moved and will show the difference at that points value. Where do the sum curves get closer? What happens as the number of terms increases near the point ? What happens as the number of terms increases at the point ?
Generalized Geometric Power Series
The Geometric Power Series can be generalized by introducing some constants to the terms. The new series is created by replacing with where are constants. The applet below illustrates this General Geometric Power Series as above with the ability to vary the constants with sliders. Note how the curves change as each constant is varied. Where do the Sum of Terms and the Infinite Sum curves get closer with additional terms? How does the convergence range vary with each constant?