Limits

  • Limits

    Grade: High School

    A geometric series is a sum of numbers such that the ratio between consecutive terms is constant. For instance, 1/2 + 1/4 + 1/8 + … is a geometric series. In this activity, you can see a representation of several geometric series and their limits.

    Activity

    Instructions

    • Choose a geometric series in the top center by moving the slider. By default, an = 0.5n is chosen.
    • Use the controls in the bottom center to step through the terms of the series.

    3571  resetReset: returns you to term 1 at any point.

    3571 previousPrevious: moves you to the previous term.

    3571 nextNext: moves you to the next term.

    3571 step50Advance to Step 50: accelerates the animation and brings you to step 50, the last given term of all series.

    • The sum of the series will be kept track of in the top left summation statement. Click on it to show all terms, and click again when you want to hide it.
    • As you explore the different terms, try to predict the limit of the series as it approaches infinity. Use the limit slider at the far right to show your prediction, or enter the value directly into the limit box at the top right. If you think the series diverges, click the button at the bottom right. After step 50, your answer will automatically be checked.
     

    Exploration

    As you explore different geometric series, see if you can answer these questions:
    • Some terms leave "traces" instead of adding solid bars. What kind of terms do this?
    • Write an inequality to express groups of series that looks roughly the same?
    • Is there any way to predict a limit without looking at the graph?
    • Some series approach their limit very quickly. What kind of series do this?
    • When does a series diverge?
     

    Objectives and Standards

    NCTM Standards and Expectations
    • Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
    • High School (9-12)
    • Algebra
    • Geometry