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Programmed tutorial: Geometric Series: Example 1

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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.

Test whether the series

is a geometric series. If it is, find its sum.

What property characterizes a geometric series?

General Contents

Detailed Contents

A common ratio of a term to the preceding one.

Does the series need to start with “1”?

No, there may be a common factor present as well as a ratio.

Does the above series have a common factor?

Yes.

Rewrite the series in factored form.

Is there a ratio of a term to the preceding one involved here?

Yes.

Determine it.

The ratio is

Is the series a geometric series?

Yes.

Write the general (n^th) term of the series.

Plot the terms of the series on paper. To check your graph, click "Next".

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Write the series using the n^th term.

Plot the sequence of partial sums,

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What value does the sequence of partial sums appear to approach?

10.

How do we find the sum analytically?

We need to determine the first term, a, and the ratio r. Then

We have r. What is the first term?

a = 2

Set up the determination of the sum.

Substituting, we get

How can we simplify this?

We can do the subtraction in the denominator first.

Do that.

Rewrite this using the “÷” symbol.

What is the next step?

“Invert and multiply”

Do it.

, in agreement with our graph above.

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