Introduction to Analysis · Chapter 05

Infinite series

Adding without end. Partial sums, the one thing every convergent series must do, the geometric family everything is compared against, the tests that follow from it, and why the order of the terms can decide the answer.

n = 1 \sum \infty a_{n}

S_{N} = n = 1 \sum N a_{n}

n = 0 \sum \infty 2^{- n} = 2

\sum ∣ a_{n} ∣

lim sup n ∣ a_{n} ∣

(- 1)^{n}

n = 1 \sum \infty \frac{1}{n ^{2}}

a_{n} \to 0

Lessons

Build the foundation

05Series of non-negative termsWhen no term is negative the partial sums can only rise. A rising sequence has just two possible fates: it passes every bound, or it settles. That reduces a question about a series to a question about whether one number can be found above all the running totals, which is often far easier to answer.Read lesson →

06The comparison testsA series of non-negative terms converges exactly when its partial sums stay bounded, and the simplest way to bound them is by another series already known to converge. That is the comparison test. Its limit form removes the need for a term-by-term inequality: if two series have terms of comparable size in the long run, they live or die together, and comparability is all that most examples can be made to show.Read lesson →

07Alternating seriesEvery test so far has needed non-negative terms. The first test that does not is for series whose signs alternate strictly: if the sizes decrease to zero, the series converges, and the error after stopping is no larger than the first term left out. The proof is a picture — the partial sums step forward and back by ever smaller amounts, trapping the answer between them — and it is the nested interval property doing the work.Read lesson →

08Absolute and conditional convergenceA series with mixed signs can converge for two quite different reasons: because its terms are genuinely small, or because its positive and negative parts happen to cancel. Taking absolute values before summing tells the two apart. A series that still converges once the signs are removed is called absolutely convergent, and it is far better behaved than one that is not.Read lesson →

09RearrangementsAddition does not care about order, and for a finite sum that settles the matter. For an infinite series it does not, because the value is a limit rather than a sum, and reordering the terms reorders the running totals. When the series converges absolutely the value survives any reordering. When it converges only conditionally the value can be moved anywhere at all.Read lesson →

10The root and ratio testsBoth tests ask the same question: does the series eventually behave like a geometric one? The ratio test compares each term with the one before, the root test looks at the size of the terms directly. If either says the effective ratio settles below one, the series is squeezed under a convergent geometric series and converges absolutely. If it settles above one the terms do not even vanish. At exactly one, both tests say nothing at all.Read lesson →

11Chapter practiceThis chapter turned every question about a series into a question about a sequence, then built tools for the cases that arise most. The questions here arrive shuffled, and deciding which tool a question calls for is the part that carries over.Read lesson →

Mapping the lesson.

Infinite series

Build the foundation