Concept preview

What you will be able to do

1. 01Define the sum of an infinite series $\sum a_n$ as the limit of its sequence of PARTIAL SUMS $S_N$, and classify a series as convergent (finite limit) or divergent (no finite limit).
2. 02Distinguish the sequence of TERMS $a_n$ from the sequence of PARTIAL SUMS $S_N$, and explain why they answer two different questions.
3. 03Evaluate a telescoping series by writing a clean closed form for $S_N$ and taking the limit, and explain why $a_n\to 0$ does NOT by itself guarantee convergence.

Start with a sequence of terms $a_1,\,a_2,\,a_3,\,\ldots$ The infinite series built from them is written $\sum_{n=1}^{\infty} a_n$. To give this endless sum a precise meaning we introduce the sequence of PARTIAL SUMS. The $N$-th partial sum is what you get by stopping after $N$ terms: $S_N=a_1+a_2+\cdots+a_N=\sum_{n=1}^{N} a_n$. So $S_1=a_1$, then $S_2=a_1+a_2$, then $S_3=a_1+a_2+a_3$, and so on. Each $S_N$ is an ordinary finite sum — no infinity anywhere — and together they form their own sequence $S_1,\,S_2,\,S_3,\ldots$