Cambridge O Level Additional Mathematics · Syllabus 4037 · Series
Sum to Infinity
What is Sum to Infinity?
The finite total that the partial sums of an infinite geometric progression approach as more and more terms are added. It exists only when the modulus of the common ratio is less than one, and it then equals the first term divided by one minus the common ratio. When the modulus of the ratio is one or greater, the terms do not shrink towards zero, the partial sums never settle, and no sum to infinity exists at all.
This definition is part of the Series chapter in Cambridge O Level Additional Mathematics.
Questions students ask about Sum to Infinity
When does a geometric progression have a sum to infinity?
Only when \(\lvert r\rvert<1\); the sum is then \(S_\infty=\dfrac a{1-r}\). This test must be checked before the formula is used, not after, because \(\dfrac a{1-r}\) still returns a number for values of \(r\) for which no infinite sum actually exists. The test is on \(\lvert r\rvert\), not on the sign of \(r\): a negative ratio such as \(r=-\tfrac13\) still converges, since \(\left\lvert-\tfrac13\right\rvert=\tfrac13<1\).