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7.3. Tests for Convergence

There are many different tests that can be used to determine whether a sequence or series converges. I'll briefly state three of the most useful, with sketches of their proofs.

Bounded and increasing sequences : A sequence that always increases, but never surpasses a certain value, converges.

This amounts to a restatement of the completeness axiom for the real numbers stated on page 157, and is therefore to be interpreted not so much as a statement about sequences but as one about the real number system. In particular, it fails if interpreted as a statement about sequences confined entirely to the rational number system, as we can see from the sequence 1, 1.4, 1.41, 1.414, ... consisting of the successive decimal approximations to \(\sqrt{2}\), which does not converge to any rational-number value.

Example 79

◊ Prove that the geometric series 1 + 1/2 + 1/4 +… converges.

◊ The sequence of partial sums is increasing, since each term is positive. Each term closes half of the remaining gap separating the previous partial sum from 2, so the sum never surpasses 2. Since the partial sums are increasing and bounded, they converge to a limit.

Once we know that a particular series converges, we can also easily infer the convergence of other series whose terms get smaller faster. For example, we can be certain that if the geometric series converges, so does the series

\[\frac{1}{1} + \frac{1}{1\times 2} + \frac{1}{1\times 2 \times 3} +... ,\]

whose terms get smaller faster than any base raised to the power n.

Alternating series with terms approaching zero : If the terms of a series alternate in sign and approach zero, then the series converges.

Sketch of a proof: The even partial sums form an increasing sequence, the odd sums a decreasing one. Neither of these sequences of partial sums can be unbounded, since the difference between partial sums n and n+1 would then have to be unbounded, but this difference is simply the nth term, and the terms approach zero. Since the even partial sums are increasing and bounded, they converge to a limit, and similarly for the odd ones. The two limits must be equal, since the terms approach zero.

Example 80

◊ Prove that the series 1 - 1/2 + 1/3 - 1/4 +… converges.

◊ Its convergence follows because it is an alternating series with decreasing terms. The sum turns out to be ln 2, although the convergence of the series is so slow that an extremely large number of terms is required in order to obtain a decent approximation,

The integral test : If the terms of a series an are positive and decreasing, and f(x) is a positive and decreasing function on the real number line such that f(n)=an, then the sum of an from n=1 to ∞ converges if and only if \(\int_1^\infty f(x)dx\) does.

Sketch of proof: Since the theorem is supposed to hold for both convergence and divergence, and is also an “if and only if,” there are actually four cases to prove, of which we pick the representative one where the integral is known to converge and we want to prove convergence of the corresponding sum. The sum and the integral can be interpreted as the areas under two graphs: one like a smooth ramp and one like a staircase. Sliding the staircase half a unit to the left, it lies entirely underneath the ramp, and therefore the area under it is also finite.

Example 81

◊ Prove that the series 1 + 1/2 + 1/3+… diverges.

◊ The integral of 1/x is ln x, which diverges as x approaches infinity, so the series diverges as well.

The ratio test : If the limit \(R=\lim_{n\rightarrow\infty}|a_{n+1}/a_n|\) exists, then the sum of an converges if R<1 and diverges if R>1.

The proof can be obtained by comparing with a geometric series.

Example 82

◊ Prove that the series 1 + 1/22 + 1/33 +… converges.

R is easily proved to be 0, so the sum converges by the ratio test.

At this point it will seem like a mystery how anyone could have proved the exact results claimed for some of the “special” series, such as 1 - 1/2 + 1/3 - 1/4 +…= ln 2. Problems like these are not the main focus of the chapter, and in fact there is no well-defined toolbox of techniques that will allow any such “nice” series to be evaluated exactly. Even a relatively innocent-looking example like 1-2 + 2-2 + 3-2 +… defeated some of the best mathematicians of Europe for years (see problem 16, p. 116). It is currently unknown whether some apparently simple series such as \(\sum_{n=1}^\infty 1/(n^3\:\sin^2 n)\) converge [1].

[1] Alekseyev, “On convergence of the Flint Hills series,” arxiv.org/abs/1104.5100v1.