Convergence of Infinite Geometric Series

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Convergence of Infinite Geometric Series

An infinite geometric series converges only when $|r|<1$ , because then $r^n\to0$ . Its sum is $S_\infty=a/(1-r)$ . If $|r|\ge1$ , terms do not shrink to zero and no finite sum exists.

Worked reasoning

The common ratio is $r=-1/3$ , so $|r|<1$ .
$S_\infty=18/(1-(-1/3))$ .
$18/(4/3)=13.5$ .

Key idea

A geometric series has a finite infinite sum exactly when $|r|<1$ .

Find the sum to infinity of $18-6+2-\frac23+\cdots$ .

Which statement best captures the central mathematical idea in Convergence of Infinite Geometric Series?

A geometric series has a finite infinite sum exactly when $|r|<1$ .

Every geometric pattern has an infinite sum because a formula exists.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Convergence of Infinite Geometric Series, which move is most reliable?

Find the common ratio and test its absolute value before using $S_\infty$ .

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Convergence of Infinite Geometric Series?

Every geometric pattern has an infinite sum because a formula exists.

A geometric series has a finite infinite sum exactly when $|r|<1$ .

Find the common ratio and test its absolute value before using $S_\infty$ .

Compute a few partial sums and confirm they approach the claimed limit.

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