Fundamentals of Series

1Introduction

An operation that adds infinitely many real numbers at once is not defined directly by finite addition. The value of an infinite series is defined as the limit of a sequence of partial sums, each containing only finitely many terms.

This lecture concerns a real sequence $(a_n{)}_{n\ge 1}$(a_n)_{n\geq1}. It formulates convergence of a series by the $\epsilon$\varepsilon-$N$N criterion, proves a necessary condition and demonstrates why its converse fails, and then classifies every real geometric series.

2Partial Sums and the Definition of a Series

For $N\ge 1$N\geq1, define the $N$Nth partial sum by

The notation $\sum _{n=1}^{\mathrm{\infty }}{a}_n$\sum_{n=1}^{\infty}a_n is interpreted through the sequence of partial sums $(S_N{)}_{N\ge 1}$(S_N)_{N\geq1}.

The series {converges} to $S\in \mathbb{R}$S\in\mathbb R if $S_N\to S$S_N\to S, in which case we write

Equivalently,

The series {diverges} if its partial sums do not converge to a finite real number. Before convergence is established, the notation $\sum _{n=1}^{\mathrm{\infty }}{a}_n$\sum_{n=1}^{\infty}a_n does not denote a real number in the same manner as a finite sum.

3A Necessary Condition for Convergence

3.1Theorem

If $\sum _{n=1}^{\mathrm{\infty }}{a}_n$\sum_{n=1}^{\infty}a_n converges, then $a_n\to 0$a_n\to0.

3.2Proof

To handle the lower endpoint uniformly, define the empty sum by $S_0=0$S_0=0. Then, for every $n\ge 1$n\geq1,

Suppose $S_n\to S$S_n\to S. Given any $\epsilon >0$\varepsilon>0, there is a $K\ge 1$K\geq1 such that $m\ge K$m\geq K implies $|S_m-S|<\epsilon /2$|S_m-S|<\varepsilon/2. If $n\ge K+1$n\geq K+1, then both $n\ge K$n\geq K and $n-1\ge K$n-1\geq K, so

Therefore $a_n\to 0$a_n\to0. $\mathrm{\square }$\square

By contraposition, if $a_n$a_n does not tend to zero, then $\sum a_n$\sum a_n diverges. The condition $a_n\to 0$a_n\to0, however, is not sufficient.

4Counterexample: The Harmonic Series

In the harmonic series

the terms satisfy $1/n\to 0$1/n\to0, but the series diverges. Indeed, for $m\ge 1$m\geq1, group the $2^{m-1}$2^{m-1} terms whose indices satisfy $2^{m-1}<n\le 2^m$2^{m-1}<n\leq2^m. Every such index satisfies $n\le 2^m$n\leq2^m, so every term is at least $1/2^m$1/2^m, and

Consequently, with no overlap at the endpoints, for $k\ge 1$k\geq1,

The right-hand side is unbounded as $k\to \mathrm{\infty }$k\to\infty. Since $S_N$S_N is increasing, in fact $S_N\to +\mathrm{\infty }$S_N\to+\infty. Thus the harmonic series diverges, proving that $a_n\to 0$a_n\to0 alone cannot guarantee convergence of a series.

5Complete Classification of Geometric Series

Let $a,r\in \mathbb{R}$a,r\in\mathbb R and consider

For $N\ge 0$N\geq0, put $T_N=\sum _{n=0}^{N}a{r}^n$T_N=\sum_{n=0}^{N}ar^n.

• If $a=0$a=0, then $T_N=0$T_N=0 for every $r$r, so the series converges to $0$0.
• If $a\ne 0$a\ne0 and $r\ne 1$r\ne1, then $(1-r)T_N=a(1-r^{N+1})$(1-r)T_N=a(1-r^{N+1}), and hence

When $\left|r\right|<1$|r|<1, we have $r^{N+1}\to 0$r^{N+1}\to0, and therefore

• If $a\ne 0$a\ne0 and $r=1$r=1, then $T_N=a(N+1)$T_N=a(N+1), which diverges to $+\mathrm{\infty }$+\infty for $a>0$a>0 and to $-\mathrm{\infty }$-\infty for $a<0$a<0.
• If $a\ne 0$a\ne0 and $r=-1$r=-1, then $T_N$T_N oscillates as $a,0,a,0,\dots$a,0,a,0,\ldots and diverges.
• If $a\ne 0$a\ne0 and $\left|r\right|>1$|r|>1, then the term $a{r}^n$ar^n does not tend to zero, so the necessary condition proves divergence. For $r>1$r>1, the partial sums tend to $+\mathrm{\infty }$+\infty if $a>0$a>0 and to $-\mathrm{\infty }$-\infty if $a<0$a<0; for $r<-1$r<-1, their signs alternate while their absolute values grow.

Thus, when $a\ne 0$a\ne0, the geometric series converges if and only if $\left|r\right|<1$|r|<1. The case $a=0$a=0 is an exception that converges for every $r$r.

6Decision Procedure

• First define the partial sums and ask whether that sequence has a limit.
• If the general term does not tend to zero, conclude immediately from the necessary condition that the series diverges.
• If the general term tends to zero, no conclusion follows yet; analyze the partial sums or apply an appropriate convergence test.
• For a geometric series, first separate $a=0$a=0; when $a\ne 0$a\ne0, determine whether $\left|r\right|<1$|r|<1.

7Scope

This lecture develops the definition of convergence and its first necessary condition. Comparison tests for positive-term series, absolute convergence, alternating series, and power series require additional theorems. Changing finitely many terms does not change whether a series converges or diverges, although in the convergent case it changes the sum by the corresponding finite amount.

8Basic Formulas

9Summary

• Convergence of an infinite series means convergence of its sequence of partial sums.
• The condition $a_n\to 0$a_n\to0 is necessary but not sufficient, as the harmonic series demonstrates.
• After the zero series is separated, a geometric series converges exactly when $\left|r\right|<1$|r|<1.

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