Series and Convergence Tests

A series adds infinitely many numbers. The sum is given meaning by the sequence of finite partial sums, so the whole theory of sequences applies to a series through that sequence. The motivation for building analysis rigorously was largely to understand series, since solutions of differential equations and expansions of functions arrive as series.

Series and partial sums

The notation ${\sum }_{n=1}^{\infty }{x}_n$ does double duty: it names the series and, when convergence holds, the limit of the partial sums. Context decides which. The index may start anywhere convenient; ${\sum }_{n=0}^{\infty }{r}^n$ and ${\sum }_{n=1}^{\infty }{r}^{n-1}$ denote the same series.

The geometric series

The one family that can be summed in closed form is the geometric series, and it underlies most convergence tests.

If $|r|\ge 1$ the terms $r^n$ do not go to zero, and the series diverges. The picture is a unit length repeatedly halved (for $r=1/2$): the pieces shrink fast enough that their total is finite.

A second family collapses because consecutive terms cancel: whenever $x_n=c_n-c_{n+1}$ for a convergent sequence $\{c_n\}$, the series telescopes and sums to $c_1-{\lim }_n{c}_n$.

The correspondence runs both ways: the terms recover from the partial sums as first differences, $x_n=s_n-s_{n-1}$.

Tails, the Cauchy criterion, and linearity

Three properties transfer directly from sequences to series.

• Tails. For any $M$, the series ${\sum }_{n=1}^{\infty }{x}_n$ and ${\sum }_{n=M}^{\infty }{x}_n$ converge together, differing only by the fixed finite sum ${\sum }_{n=1}^{M-1}{x}_n$. Convergence is a property of the tail.⁴
• Cauchy criterion. Since a real sequence converges iff it is Cauchy, a series converges iff its partial sums are Cauchy: for every $ϵ>0$ there is $M$ with $$\left|\sum _{i=n+1}^k{x}_i\right|<ϵ\text{for all }k>n\ge M.$$ This says every sufficiently late block of consecutive terms has small sum.⁵
• Linearity. If $\sum x_n$ and $\sum y_n$ converge and $\alpha \in \mathbb{R}$, then $\sum \alpha x_n=\alpha \sum x_n$ and $\sum (x_n+y_n)=\sum x_n+\sum y_n$. Each follows by taking limits of the corresponding identity for partial sums.⁶

The n-th term test

The Cauchy criterion with $k=n+1$ isolates a single term, yielding a fast necessary condition for convergence.

The test is a divergence test: if the terms do not go to zero, the series diverges. It gives at once that ${\sum }_{n=0}^{\infty }{r}^n$ diverges for $|r|\ge 1$. The converse is false, as the harmonic series shows.

The harmonic series diverges

The terms tend to zero, but not fast enough: their partial sums outgrow every bound, albeit slowly. Deciding whether a nonnegative series converges is therefore a question of how fast its terms decay, and the comparison test answers it.

Nonnegative series and the comparison test

When every term is nonnegative, the partial sums are monotone increasing, so by the monotone convergence theorem the series converges exactly when they are bounded above.

This reduces convergence of a nonnegative series to one inequality, and comparing against a series already understood settles it.

The p-series

The benchmark family for the comparison test is the $p$-series.

The direction of the block estimate is the whole difference: bounding blocks from below shows divergence, from above shows convergence. Neither test names the sum. The value $\sum \frac{1}{n^2}=\frac{{\pi }^2}{6}$ is a separate and much harder computation, and $\sum \frac{1}{n^p}$ as a function of $p$ is the Riemann zeta function, the subject of a famous open problem.

Summary

A series converges by definition when its partial sums do, so sequences supply all the theory: the Cauchy criterion, linearity, and tail-independence carry over unchanged. The $n$-th term test rejects any series whose terms miss zero, but passing it guarantees nothing, as the harmonic series shows. For nonnegative terms, convergence is boundedness of the partial sums, and the comparison test against the geometric and $p$-series decides most cases at sight. The comparison test requires nonnegative terms; series with mixed signs need absolute convergence and the ratio, root, and alternating-series tests.