An infinite sum is not a mystical elongated plus sign. It is a limit of partial sums. If those partial sums approach a number, we say the series converges and we assign that number as the sum. If they wander or blow up, the series diverges — and writing a “total” is meaningless in the usual sense.
Sequences first
A sequence $(a_n)$ is an ordered list of numbers. It converges to $L$ when terms eventually stay arbitrarily close to $L$. Series piggyback on this idea: given terms $a_n$, form $s_N=a_1+\cdots+a_N$. The series $\sum a_n$ converges exactly when the sequence $(s_N)$ converges.
Classic behaviors
- Geometric series $\sum r^n$ converges when $|r|<1$, with sum formulas you can derive from partial sums.
- Harmonic series $\sum 1/n$ diverges despite terms going to zero — “terms → 0” is necessary but not sufficient.
- Alternating examples can converge conditionally; absolute convergence is a stronger, rearrangement-friendly property.
Example Partial sums of $1+\frac12+\frac14+\frac18+\cdots$ approach $2$. Partial sums of $1+\frac12+\frac13+\frac14+\cdots$ grow without bound (slowly), so there is no finite sum.
These ideas sit at the hinge between calculus technique and analysis rigor — see Real analysis & proof habits and Calculus foundations.
Citations & further reading
- OpenStax Calculus Volume 2, sequences and series. openstax.org
- Abbott, Understanding Analysis — sequences and series chapters. Springer
- Paul’s Online Math Notes, Series & Sequences. tutorial.math.lamar.edu
- MIT OCW single-variable calculus — infinite series units. ocw.mit.edu