Series, Partial Sums, and Geometric Series

A series asks about accumulation

A sequence is a list: $a_1,a_2,a_3,\ldots$. A series is the indicated sum of a sequence’s terms:

$$\sum_{n=1}^{\infty}a_n=a_1+a_2+a_3+\cdots.$$

The infinity symbol does not instruct us to perform an endless final addition. It defines a new sequence, the sequence of partial sums. The $N$th partial sum is

$$S_N=\sum_{n=1}^{N}a_n.$$

The series converges precisely when the partial sums $S_N$ approach a finite number. That limit, if it exists, is called the sum of the series. Keeping the terms $(a_n)$ separate from the partial sums $(S_N)$ prevents a central confusion: terms can approach zero while the running total behaves in many different ways.

Finite sums reveal the mechanism

Consider the repeated-halving process

$$1+\frac12+\frac14+\frac18+\cdots.$$

Its first partial sums are $1$, $1.5$, $1.75$, and $1.875$. Each new term fills half of the distance remaining to $2$. The total rises but never exceeds $2$, and it can be made arbitrarily close to $2$ by adding enough terms. Thus the infinite series has the finite sum $2$. This is not a contradiction: infinitely many positive additions can have a bounded total when the additions shrink in the right way.

For any geometric partial sum with first term $a$ and common ratio $r\ne1$,

$$S_N=a+ar+ar^2+\cdots+ar^{N-1} =a\frac{1-r^N}{1-r}.$$

The formula follows by multiplying the sum by $r$ and subtracting aligned terms. The cancellation is structural, so it only applies when every term is obtained from the preceding term by the same multiplier. A pattern that merely “looks like it gets smaller” is not enough to justify the geometric formula.

The ratio decides a geometric series

Letting $N\to\infty$ in the geometric partial-sum formula requires understanding $r^N$. If $|r|<1$, then $r^N\to0$ and

$$\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}.$$

If $|r|\ge1$, the geometric series diverges. When $r=1$, partial sums grow by the same nonzero amount. When $r=-1$, they alternate between values. When $|r|>1$, the terms themselves do not approach zero. These are different visible behaviors, but each prevents the partial sums from approaching one finite total.

For example,

$$\sum_{n=0}^{\infty}3\left(-\frac{2}{5}\right)^n =\frac{3}{1-(-2/5)}=\frac{15}{7}.$$

The negative ratio causes alternating terms, but its magnitude is less than one, so their effects shrink. The sign does not make the series diverge by itself. The condition is on absolute value: $|-2/5|<1$.

The term test is a necessary check

If a series $\sum a_n$ converges, then its terms must satisfy $a_n\to0$. This is called the divergence test, or the $n$th-term test for divergence. It is fast: since $\lim_{n\to\infty}(n+1)/n=1$, the series $\sum(n+1)/n$ diverges immediately because its terms do not approach zero.

The reverse statement is false. The harmonic series

$$1+\frac12+\frac13+\frac14+\cdots$$

has terms approaching zero but still diverges. Terms becoming small is necessary because otherwise a running total cannot settle, but it is not sufficient because there may be too many small contributions. This logical boundary explains why calculus develops several additional convergence tests.

It helps to separate the two questions explicitly. The question “do the terms vanish?” is answered by the limit of $a_n$. The question “does the accumulated total settle?” is answered by the limit of $S_N$. In a convergent series these answers are connected, but they are never interchangeable. Recording both sequences in a table for the first few indices makes the distinction concrete: one column contains what is added next, while the other contains the total after that addition.

Finite arithmetic can also mislead if it is treated as a proof of an infinite claim. A calculator may report that the first thousand harmonic partial sums are only modest in size, but “modest so far” does not mean bounded forever. Conversely, a convergent series may approach its sum so slowly that a short table looks inconclusive. The definition always returns to the same target: can all sufficiently late partial sums be forced inside any desired tolerance of one finite number? Later tests are tools for answering that question without needing a closed form for $S_N$.

A setup routine for every series

First name the term $a_n$ and the partial sum $S_N$. Next look for a constant ratio, but verify it by computing $a_{n+1}/a_n$ rather than guessing from a few terms. If the series is geometric, state $a$, $r$, the condition $|r|<1$, and then use the sum formula. If it is not geometric, do not force the formula; test the terms for the necessary zero limit and choose a later method based on the series’ structure.

Pay special attention to the index at which a sum begins. A formula written with $n=0$ has first term $a$, while one written with $n=1$ may have first term $ar$ or an entirely different expression. Rewrite the first two or three actual terms before identifying $a$ and $r$. That small check catches the common error of using the displayed coefficient as the first term when the exponent or index shifts it. It also makes your answer auditable by another reader without asking them to reconstruct the pattern from notation alone.

For retrieval, decide whether $\sum_{n=0}^{\infty}5(3/4)^n$ converges and find its sum. Then compare it with $\sum_{n=0}^{\infty}5(4/3)^n$: describe the term behavior and explain why the geometric formula may not be used to claim a finite sum. The next lesson develops two broad tests—the divergence test and the integral test—for series that are not geometrically organized.