Which a, Which r, and Does It Converge

You'll learnto derive the closed form for a geometric partial sum and see where the convergence condition comes from, to take the numerator from the first term actually present rather than from the coefficient out front, to find the ratio by dividing consecutive terms whatever the notation is doing, to recognise the formula returning a confident answer to a series it does not apply to, and to evaluate a repeating decimal as the geometric series it is.

A geometric series is the first one whose sum you can write down, and the fraction that does it is three decisions compressed into one expression. Does the series converge at all, which term is the numerator, and which number is the ratio. Each of the three fails independently of the other two, and a wrong answer looks exactly like a right one.

8 STEPS · 6 QUICK CHECKS · THREE DECISIONS IN ONE FRACTION · v1

substitute the lower limit · divide consecutive terms · check |r| first

Before you start

What you're looking at

One index panel. The horizontal axis is n and takes whole-number values only. Pink stems standing on the axis are the terms, violet dots above them are the running totals, and a lime rule is a limit. Teal is whatever is being compared against the violet sequence — a second series, or a rival evaluation of the same one.

Every wrong answer is drawn as a height

The rules at 6, 3 and 1.5 in step 3 are all correct sums — of three different series. Nothing on the panel is labelled wrong because it is badly computed; they are wrong because they answer a question that was not asked, and seeing them at three different heights is the point.

Watch for

In step 2 the partial sums leave the top of the panel while the formula's answer sits below the axis at −1. Both are on screen at once. A negative number under a rising sum of positive terms is the clearest signal you will get that a formula has been applied outside its conditions.

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