An infinite sum means a finite value approached by the partial sums S_1,S_2,, not a sum with a very large chosen number of terms. For a nonzero first term and r 1, S_n= a1-r- arn1-r. When |r|<1, repeated multiplication makes rn approach zero; therefore S_n approaches S_infinity= a1-r. For 8,4,2,1,, partial sums 8,12,14,15, approach 16. The remaining amount after n terms is 16(12)n, which approaches zero. A negative ratio can converge: 6,-3, 32, has r=- 12, so the partial sums oscillate towards 61+ 12=4. For r=1, nonzero equal terms keep changing the total. For r=-1, the totals alternate instead of approaching one number. For |r|>1, the terms do not approach zero. These nonzero GPs have no finite sum to infinity. The convergence condition -1<r<1 must be checked before using the supplied sum formula. An algebraic value from a1-r is not evidence that the series converges.

An infinite sum means a finite value approached by the partial sums \(S_1,S_2,\ldots\), not a sum with a very large chosen number of terms.

For a nonzero first term and \(r\ne1\), \(S_n=\dfrac a{1-r}-\dfrac{ar^n}{1-r}\). When \(|r|<1\), repeated multiplication makes \(r^n\) approach zero; therefore \(S_n\) approaches \(S_\infty=\dfrac a{1-r}\).

For \(8,4,2,1,\ldots\), partial sums \(8,12,14,15,\ldots\) approach \(16\). The remaining amount after \(n\) terms is \(16\left(\dfrac12\right)^n\), which approaches zero.

A negative ratio can converge: \(6,-3,\dfrac32,\ldots\) has \(r=-\dfrac12\), so the partial sums oscillate towards \(\dfrac6{1+\frac12}=4\).

For \(r=1\), nonzero equal terms keep changing the total. For \(r=-1\), the totals alternate instead of approaching one number. For \(|r|>1\), the terms do not approach zero. These nonzero GPs have no finite sum to infinity.

The convergence condition \(-1<r<1\) must be checked before using the supplied sum formula. An algebraic value from \(\dfrac a{1-r}\) is not evidence that the series converges.

When an infinite sum exists

When an infinite sum exists