Math With AmarA C A D E M Y
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High school · Sequences and series

Add a shrinking geometric series

Watch finite partial sums fill a bar while a strictly positive remainder stays visible in the numbers.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Add a shrinking geometric series. Terms added: 1. Partial sum: 2. Infinite total: 4. Positive remainder: 2Adding term 1 of 12Total = 40S∞First 8 terms, relative to a:aarar²ar³ar⁴ar⁵ar⁶ar⁷Pale bar = all terms still to come.
This model covers positive a and 0 < r < 1 only. The bar is normalized to the current infinite total, so its scale changes when a or r changes. Playback adds whole terms; the lower chart shows the first eight term magnitudes relative to a.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Terms added
1
Partial sum
2
Infinite total
4
Positive remainder
2

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From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

A geometric series multiplies each term by the same ratio. With a positive first term a and 0 < r < 1, the terms shrink and the partial sums approach a/(1 − r). Infinitely many positive terms can have a finite total, but no finite partial sum equals that total.

A relationship to keep

Sₙ = a(1 − rⁿ)/(1 − r); S∞ − Sₙ = arⁿ/(1 − r)

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Add one term at a time

    Each colored bar segment represents a, ar, ar², and so on. Playback adds up to twelve terms. Segment width is proportional to that term as a fraction of the infinite total.

  2. STEP 2

    Keep track of the missing tail

    The pale unfilled portion is the sum of all terms still to come. It shrinks as n grows. For small ratios it may become narrower than a screen pixel, but the displayed remainder remains positive.

  3. STEP 3

    Compare convergence speeds

    Set a larger r and replay. Terms shrink more slowly and twelve terms leave a larger fraction of the total missing. Changing a scales every term and the total together, so the fraction-filled picture stays the same.

Your turn to explain

Make a prediction. Test your reasoning.

With a = 2, r = 1/2, and n = 3, what are the partial sum, infinite total, and remainder?

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The first three terms are 2, 1, and 1/2. Their sum is 3.5, the infinite total is 4, and the remainder is 0.5.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 11 · Algebra

Add repeated multiplicative contributions efficiently

Derive a finite geometric-sum formula by subtraction.

Grade 12 · Algebra

Control the tail of an infinite geometric sum

Check convergence and quantify how much a finite truncation misses.