A-level · Interactive explorer

Series Studio

Explore sigma notation, sum formulas, convergence and logarithmic cancellation in eight interactive lessons.

Predict, test and explain

Step 1 of 3

Predict

What is the sum to infinity of 20(½)⁴ + 20(½)⁵ + …?

Eight lessons with sliders, graphs, step-by-step examples and practice. Explore here, or open the full-screen studio.

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Choose the investigation above. Predict what will change before you move a control.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

What is the sum to infinity of 20(½)⁴ + 20(½)⁵ + …?

a=20(1/2)4a=20(1/2)^4

Hint

Substitute the lower index to find the first included term.

Reveal answer

The first included term is 5/4, so divide 5/4 by 1 − 1/2.

5/25/2

Build an explanation

Hint 1

Substitute the lower index to find the first included term.

Hint 2

Check the magnitude of the ratio is less than 1.

Hint 3

The coefficient is not necessarily the first included term.

Worked example

This example uses fixed values, separate from the diagram controls.

Sum 20(½)ʳ from r = 4 to infinity.

  1. The first included term is 20/16 = 5/4.
  2. The ratio is 1/2, so the series converges.
  3. Calculate (5/4)/(1 − 1/2).

Answer: 5/2.

Connect this idea

Watch out: Using 20 as the first term ignores the lower index.