Finite sums have a comforting stability: adding $a_1,\ldots,a_n$ gives a number, and adding a few more small terms usually feels like a small correction. Infinite sums test that intuition. The expression \begin{align*} 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots \end{align*} looks harmless because the individual terms tend to $0$, yet the accumulated total grows beyond every fixed bound. A series is the mathematical device that separates two questions that finite notation hides: do the partial sums approach a limit, and if they do, how stable is that limit under estimates, rearrangements, and limiting processes? The central idea is to stop thinking of an infinite sum as a completed addition and instead study the sequence of finite approximations. If the partial sums settle down, the series converges. If they do not, the formal infinite expression has no ordinary value. This shift is small in notation and large in consequence: most of the theory of series is a theory of controlling tails. [example: Harmonic Series as a First Failure] For the harmonic series, let \begin{align*} s_N=\sum_{k=1}^N \frac{1}{k}. \end{align*} The individual terms tend to zero, but the partial sums are not bounded. For each integer $m\ge 0$, \begin{align*} \sum_{k=2^m+1}^{2^{m+1}} \frac{1}{k} \ge \sum_{k=2^m+1}^{2^{m+1}} \frac{1}{2^{m+1}} = \frac{2^m}{2^{m+1}} = \frac{1}{2}. \end{align*} Hence along the subsequence $N=2^M$, \begin{align*} s_{2^M} &=1+\sum_{m=0}^{M-1}\sum_{k=2^m+1}^{2^{m+1}}\frac{1}{k}\\ &\ge 1+\frac{M}{2}. \end{align*} The right-hand side tends to infinity, so $(s_N)$ cannot converge. This is the basic warning that $a_n\to0$ is necessary for convergence of $\sum a_n$, but it is not sufficient. [/example] This example explains why the condition $a_n \to 0$ cannot be the whole story. It is necessary, but it does not measure how much mass remains in the tail. The rest of the chapter develops increasingly sharp ways of measuring that tail: direct computation, comparison, cancellation, and analytic continuation through [power series](/page/Power%20Series). Before defining the infinite object, we need to isolate the finite approximations. This keeps the discussion tied to [sequences](/page/Sequence) and [limits](/page/Limit), where convergence already has a precise meaning. [definition: Sequence of Partial Sums] Let $(a_n)_{n=1}^\infty$ be a sequence in $\mathbb{R}$ or $\mathbb{C}$. The sequence of partial sums associated to $(a_n)$ is the sequence $(s_N)_{N=1}^\infty$ defined by \begin{align*} s_N = \sum_{n=1}^N a_n. \end{align*} [/definition] The notation $\sum_{n=1}^\infty a_n$ suggests adding infinitely many terms at once, but the rigorous object is built from finite sums. Once the partial sums have been named, the infinite sum is not a new algebraic operation. It is a limit question about the sequence $(s_N)$. [definition: Series] Let $(a_n)_{n=1}^\infty$ be a sequence in $\mathbb{R}$ or $\mathbb{C}$, and let $(s_N)$ be its sequence of partial sums. The series with terms $a_n$ is the formal expression \begin{align*} \sum_{n=1}^\infty a_n. \end{align*} It converges to $s$ if \begin{align*} \lim_{N \to \infty} s_N = s. \end{align*} It diverges if the sequence $(s_N)$ does not converge. [/definition] The value of a convergent series is therefore defined by \begin{align*} \sum_{n=1}^\infty a_n = \lim_{N \to \infty} \sum_{n=1}^N a_n. \end{align*} This convention matters: every theorem about series is a theorem about limits of partial sums. The geometric series is the model case because its partial sums can be computed exactly. It gives the first reliable test: repeated multiplication by a fixed ratio converges precisely when the ratio has modulus less than $1$. [example: Geometric Series] Fix $r\in\mathbb{C}$ and write \begin{align*} s_N=\sum_{n=0}^N r^n. \end{align*} If $r\ne 1$, multiplying by $1-r$ gives the telescoping identity \begin{align*} (1-r)s_N &= (1+r+\cdots+r^N)-(r+r^2+\cdots+r^{N+1})\\ &=1-r^{N+1}, \end{align*} and therefore \begin{align*} s_N=\frac{1-r^{N+1}}{1-r}. \end{align*} When $|r|<1$, the term $r^{N+1}$ tends to $0$, so \begin{align*} \sum_{n=0}^{\infty}r^n=\frac{1}{1-r}. \end{align*} If $r=1$, then $s_N=N+1$, so the series diverges. If $|r|\ge 1$ and $r\ne1$, then $|r^n|=|r|^n$ does not tend to $0$; the terms of the series fail the term test, so the series diverges. [/example] This calculation is the seed for many later tests. Whenever a complicated series can be bounded above or below by a geometric series, convergence or divergence often follows. **The Cauchy Viewpoint.** A series is often understood by ignoring its first few terms and studying what remains. The initial terms affect the numerical value, but convergence is controlled by the eventual behaviour. [definition: Tail of a Series] Let $\sum_{n=1}^\infty a_n$ be a series. For $N \in \mathbb{N}$, the $N$-th tail is the series \begin{align*} \sum_{n=N+1}^\infty a_n. \end{align*} [/definition] Tails are the working object in estimates. Saying that a series converges is the same as saying that sufficiently far-out tails become small in total, not merely small term by term. Computing the exact sum of a series is rare. More often, we need to know whether a sum exists before knowing what it equals. The [Cauchy](/page/Cauchy%20Sequence) viewpoint answers this by comparing partial sums to each other instead of comparing them to a proposed limit. [quotetheorem:3825] This theorem says that the total contribution of every sufficiently late finite block must be small. It is stronger than asking each individual term to be small, because a large number of small terms may still add up to something significant. A first consequence is the most basic necessary condition for convergence. It is often called the term test, but its real role is diagnostic: it detects divergence, never convergence. [quotetheorem:3826] The harmonic series shows why the converse fails. The term test is a gate at the entrance of the subject: failing it ends the discussion, passing it starts the real work. [example: A Series That Fails the Term Test] Consider \begin{align*} \sum_{n=1}^{\infty}(-1)^n. \end{align*} The terms do not tend to zero, since the subsequences $((-1)^{2m})_{m\ge1}$ and $((-1)^{2m-1})_{m\ge1}$ are constantly $1$ and $-1$, respectively. The partial sums show the same obstruction explicitly: \begin{align*} s_{2m}=0,\qquad s_{2m-1}=-1. \end{align*} Thus the sequence of partial sums has two different subsequential limits, so the series diverges. [/example] Convergence tests need to ignore harmless initial irregularities, since most estimates only become true after some point. The permanence principle we need says that changing finitely many terms can alter the numerical sum but cannot alter convergence itself. [quotetheorem:3827] This is why convergence tests usually state conditions for all sufficiently large $n$. A finite number of exceptional terms may change the sum, but they cannot decide whether a limiting sum exists. **Positive Series and Comparison.** When every term is nonnegative, cancellation disappears. The partial sums then form an increasing [monotone sequence](/page/Monotone%20Sequence), so the only possible obstruction to convergence is unbounded growth. This turns many convergence questions into bounding questions. [definition: Positive Series] A positive series is a series $\sum_{n=1}^\infty a_n$ with $a_n \ge 0$ for every $n \in \mathbb{N}$. [/definition] For positive series, convergence means the partial sums are bounded above. To use this monotonicity in practice, we need a boundedness criterion that replaces finding the exact sum by finding any finite upper bound. [quotetheorem:3828] Most positive series are not summed exactly, so we need a way to transfer convergence from a known benchmark to an unknown tail. The comparison principle captures the simple idea that a smaller nonnegative tail cannot diverge if a larger nonnegative tail has finite total mass. [quotetheorem:173] Comparison is most useful when paired with a family whose convergence is fully understood. The basic benchmark we need is a one-parameter family whose tail size changes exactly at one visible threshold. [definition: P-Series] For $p \in \mathbb{R}$, the $p$-series is the series \begin{align*} \sum_{n=1}^\infty \frac{1}{n^p}. \end{align*} [/definition] The threshold at $p=1$ is one of the central landmarks of elementary analysis. Above it, the tail has finite mass. At and below it, the tail is too large. [quotetheorem:3834] This theorem turns rough asymptotic information into convergence information. A series that behaves like $1/n^{3/2}$ should converge; one that behaves like $1/n$ should not. [example: Comparison with a P-Series] Consider \begin{align*} \sum_{n=1}^{\infty}\frac{3n+1}{n^3+n}. \end{align*} Every term is nonnegative. For $n\ge1$, \begin{align*} 0\le \frac{3n+1}{n^3+n} \le \frac{4n}{n^3+n} \le \frac{4n}{n^3} =\frac{4}{n^2}. \end{align*} The comparison series $\sum_{n=1}^{\infty}4/n^2$ converges, because it is a constant multiple of a $p$-series with $p=2>1$. Direct comparison therefore gives convergence of the original series. [/example] Sometimes termwise inequalities are awkward, but the quotient of two positive sequences has a limit. To use asymptotic equivalence as a convergence tool, we need a comparison test that reads the eventual quotient rather than a pointwise bound. [quotetheorem:3829] A different comparison is available when the term $a_n$ comes from sampling a decreasing function. We need this test because areas under a graph can estimate sums even when no simple benchmark series has been guessed. [quotetheorem:196] The integral test is especially good at detecting logarithmic corrections. It shows, for example, that adding a power of $\log n$ to the denominator can change convergence exactly at a boundary. [example: A Logarithmic Boundary] For $q\in\mathbb{R}$, consider \begin{align*} \sum_{n=2}^{\infty}\frac{1}{n(\log n)^q}. \end{align*} The function $f(x)=1/(x(\log x)^q)$ is positive for $x>1$ and eventually decreasing: for $x$ large enough, \begin{align*} f'(x) =-\frac{\log x+q}{x^2(\log x)^{q+1}}<0. \end{align*} The integral test applies after discarding finitely many initial terms. With $u=\log x$, for $B>A>1$ we get \begin{align*} \int_A^B \frac{1}{x(\log x)^q}\,dx =\int_{\log A}^{\log B}u^{-q}\,du. \end{align*} If $q\ne1$, this equals \begin{align*} \frac{(\log B)^{1-q}-(\log A)^{1-q}}{1-q}, \end{align*} which has a finite limit as $B\to\infty$ exactly when $q>1$. If $q=1$, the integral is $\log(\log B)-\log(\log A)$ and diverges. Hence the series converges exactly for $q>1$. [/example] Many series contain powers, factorials, or exponential growth, where direct comparison is not obvious from the terms themselves. We need tests that compare successive or exponential scales directly to a geometric series without first guessing the comparison sequence. [quotetheorem:174] When the $n$-th term already carries an $n$-th power, ratios can hide the relevant scale. The root test is needed to measure exponential size directly and decide convergence from that scale. [quotetheorem:175] The inconclusive case $L=1$ is not a defect in wording. The harmonic series and the convergent series $\sum 1/n^2$ both have ratio and root limits equal to $1$, so no test based only on first-order exponential scale can distinguish them. **Absolute and Conditional Convergence.** Signs and complex phases can make a series converge by cancellation. We need to separate convergence caused by genuine summability of magnitudes from convergence caused only by balancing positive and negative contributions. [definition: Absolute Convergence] A series $\sum_{n=1}^\infty a_n$ in $\mathbb{R}$ or $\mathbb{C}$ is absolutely convergent if the positive series \begin{align*} \sum_{n=1}^\infty |a_n| \end{align*} converges. [/definition] Absolute convergence is the robust form of convergence. Finite total mass of magnitudes forces ordinary convergence of the original numerical series, because the tails of $\sum |a_n|$ control the tails of $\sum a_n$ by the triangle inequality. The converse fails, and the failure is not a minor exception. We need a separate name for convergence that depends on cancellation, because order and summation methods begin to matter precisely in that regime. [definition: Conditional Convergence] A series $\sum_{n=1}^\infty a_n$ in $\mathbb{R}$ or $\mathbb{C}$ is conditionally convergent if $\sum_{n=1}^\infty a_n$ converges and $\sum_{n=1}^\infty |a_n|$ diverges. [/definition] The alternating harmonic series is the standard example. To prove convergence caused by alternating cancellation rather than absolute summability, we need a test that uses decreasing magnitudes and opposite signs to trap the limit from both sides. [quotetheorem:177] This theorem gives more than convergence. It gives an error estimate, and that estimate is often the practical reason alternating series are useful. [example: Alternating Harmonic Series] For the alternating harmonic series, \begin{align*} \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}, \end{align*} set $b_n=1/n$. Then $b_n>0$, the sequence $(b_n)$ is decreasing, and $b_n\to0$. The [alternating series test](/theorems/177) therefore gives convergence. The absolute series is \begin{align*} \sum_{n=1}^{\infty}\left|\frac{(-1)^{n+1}}{n}\right| =\sum_{n=1}^{\infty}\frac{1}{n}, \end{align*} which is the divergent harmonic series. Thus the convergence is conditional, not absolute. [/example] Conditional convergence warns us that infinite sums do not always behave like finite sums. To know when order can safely be ignored, we need a rearrangement theorem that separates the robust absolute case from the order-sensitive conditional case. [quotetheorem:178] This is the algebraic payoff of absolute convergence: the infinite addition becomes order-independent in the same way finite addition is. **Power Series.** A numerical series has fixed coefficients, but analysis often needs sums whose terms depend on a variable. A power series is the object needed to turn a coefficient sequence into a function while retaining control through the theory of series. [definition: Power Series] Let $(c_n)_{n=0}^\infty$ be a sequence in $\mathbb{R}$ and let $x_0 \in \mathbb{R}$. A real power series centered at $x_0$ is a series of the form \begin{align*} \sum_{n=0}^\infty c_n (x-x_0)^n. \end{align*} Its associated function is the map \begin{align*} f: D &\to \mathbb{R} \\ x &\mapsto \sum_{n=0}^\infty c_n (x-x_0)^n, \end{align*} where $D \subset \mathbb{R}$ is the set of real numbers for which the series converges. Let $(c_n)_{n=0}^\infty$ be a sequence in $\mathbb{C}$ and let $z_0 \in \mathbb{C}$. A complex power series centered at $z_0$ is a series of the form \begin{align*} \sum_{n=0}^\infty c_n (z-z_0)^n. \end{align*} Its associated function is the map \begin{align*} f: D &\to \mathbb{C} \\ z &\mapsto \sum_{n=0}^\infty c_n (z-z_0)^n, \end{align*} where $D \subset \mathbb{C}$ is the set of complex numbers for which the series converges. [/definition] The remarkable feature of a power series is that its convergence set is organized by a single number. We need to name this number because the root test makes it the basic geometric invariant: inside it the series converges absolutely, outside it diverges, and at the boundary separate arguments are needed. [definition: Radius of Convergence] Let $\sum_{n=0}^\infty c_n (z-z_0)^n$ be a complex power series. Its radius of convergence is the unique number $R \in [0,\infty]$ such that the series converges absolutely for $|z-z_0|<R$ and diverges for $|z-z_0|>R$. [/definition] For real power series the same definition applies with $|x-x_0|$ in place of $|z-z_0|$. To compute the radius from coefficients, we need a formula that extracts the exponential growth rate of the sequence $(c_n)$. [quotetheorem:203] This formula is the root test applied uniformly to the variable term $c_n(z-z_0)^n$. It packages the exponential growth rate of the coefficients into the geometry of convergence. [example: Endpoint Behaviour Is Separate] Compare the three real power series centered at $0$: \begin{align*} \sum_{n=1}^{\infty}x^n,\qquad \sum_{n=1}^{\infty}\frac{x^n}{n},\qquad \sum_{n=1}^{\infty}\frac{x^n}{n^2}. \end{align*} Their coefficient sequences satisfy \begin{align*} \limsup_{n\to\infty}|1|^{1/n} =\limsup_{n\to\infty}\left(\frac{1}{n}\right)^{1/n} =\limsup_{n\to\infty}\left(\frac{1}{n^2}\right)^{1/n} =1, \end{align*} so all three have [radius of convergence](/theorems/262) $1$. At $x=1$ they become \begin{align*} \sum 1,\qquad \sum \frac{1}{n},\qquad \sum \frac{1}{n^2}, \end{align*} so the first two diverge and the third converges. At $x=-1$ they become \begin{align*} \sum (-1)^n,\qquad \sum \frac{(-1)^n}{n},\qquad \sum \frac{(-1)^n}{n^2}. \end{align*} The first diverges, the second converges conditionally, and the third converges absolutely. The common radius controls the open interval $(-1,1)$, but the endpoints require separate tests. [/example] Inside the interval or disk of convergence, power series behave much better than arbitrary pointwise limits. They may be differentiated and integrated term by term without changing the radius of convergence, so calculus operations become algebraic operations on coefficients. This is one reason power series are central in analysis. **Rearrangement and Summation Methods.** The ordinary definition of convergence is intentionally strict. It refuses to assign a value to $1-1+1-1+\cdots$ because the partial sums oscillate. Before discussing weaker summation procedures, we need to define how changing the order of a genuinely convergent series affects its value. [definition: Rearrangement of a Series] Let $\sum_{n=1}^\infty a_n$ be a series, and let $\pi: \mathbb{N} \to \mathbb{N}$ be a bijection. The rearrangement of $\sum_{n=1}^\infty a_n$ by $\pi$ is the series \begin{align*} \sum_{n=1}^\infty a_{\pi(n)}. \end{align*} [/definition] Rearrangements expose the difference between absolute and conditional convergence. We need a precise theorem because absolute convergence makes the order irrelevant, while conditional convergence leaves the value vulnerable to the chosen order. [quotetheorem:3831] This theorem is a warning about notation. The symbol $\sum a_n$ carries an order unless absolute convergence has removed the dependence on order. A different issue arises when the partial sums do not converge but their averages do. We need to name this averaging procedure carefully, because [summability methods](/page/Summability%20Methods) are extensions with their own rules rather than replacements for convergence. [definition: Cesaro Summability] Let $\sum_{n=0}^\infty a_n$ be a series with partial sums $(s_N)_{N=0}^\infty$. The series is Cesaro summable to $s$ if \begin{align*} \lim_{N \to \infty} \frac{1}{N+1}\sum_{k=0}^N s_k = s. \end{align*} [/definition] Cesaro summability assigns a value to some oscillatory series while respecting ordinary convergence whenever ordinary convergence exists. We need the compatibility result because a summability method should not change sums that already converge in the ordinary sense. [quotetheorem:3832] The simplest example is Grandi series. Its partial sums oscillate, but their averages settle at the midpoint of the oscillation. [example: Grandi Series] The Grandi series is \begin{align*} 1-1+1-1+\cdots=\sum_{n=0}^{\infty}(-1)^n. \end{align*} Its ordinary partial sums are \begin{align*} s_N=\sum_{n=0}^N(-1)^n =\begin{cases} 1, & N\text{ even},\\ 0, & N\text{ odd}. \end{cases} \end{align*} Since $(s_N)$ oscillates between $1$ and $0$, the series diverges in the usual sense. Its Cesaro means are \begin{align*} \sigma_N =\frac{1}{N+1}\sum_{k=0}^N s_k =\frac{\lfloor N/2\rfloor+1}{N+1}, \end{align*} because $s_k=1$ exactly for even $k$. These averages tend to $1/2$, so the series is Cesaro summable to $1/2$ even though it is not convergent as an ordinary series. [/example] Another summation method approaches a boundary value through a power series. We need to name the extension procedure before using it, so that the method is not confused with the theorem saying when it agrees with ordinary convergence. [definition: Abel Summability] Let $\sum_{n=0}^\infty a_n$ be a real or complex series. The series is Abel summable to $s$ if the power series \begin{align*} \sum_{n=0}^\infty a_n r^n \end{align*} converges for every $r \in (0,1)$ and \begin{align*} \lim_{r \uparrow 1} \sum_{n=0}^\infty a_n r^n = s. \end{align*} [/definition] Abel summability is useful because it replaces an unstable boundary expression by stable values inside the interval. We need a compatibility theorem saying that this procedure agrees with ordinary convergence whenever ordinary convergence already exists. [quotetheorem:3833] Abel theorem says that damping the terms by $r^n$ and then letting $r$ approach $1$ recovers the ordinary sum when that ordinary sum exists. It is another expression of the central principle of the chapter: infinite addition is controlled through limiting processes, and each limiting process must declare its rules. **References.** Apostol, *Mathematical Analysis* (1974). Rudin, *Principles of Mathematical Analysis* (1976). Knopp, *Theory and Application of Infinite Series* (1990). Hardy, *Divergent Series* (1949).