1.3.1—Standard series sums

The arithmetic series 1+2+…+n equals n(n+1)/2. The geometric series a+ar+…+ar^{n−1} equals a(1−r^n)/(1−r) when r≠1, and tends to a/(1−r) when |r|<1.

Index the first and last terms carefully. For an infinite series, convergence requires the terms to tend to zero; a ratio with |r|≥1 does not give a finite sum.

The sum 3+6+12+24 is a=3, r=2, n=4, so it is 3(1−2^4)/(1−2)=45. The corresponding infinite series diverges because |r|>1.

Do not use the infinite formula for a finite or divergent series, and do not confuse the number of terms with the final index.