AngouriMath

Symbolics for .NET


Table of contents

AngouriMath

Navigation
Home Quick start Wiki Almanac Research What's new Try Use Angouri.org GitHub Report a bug Donate Contacts

DivergentSeries

← Back to list of classes

Description

Summary

A summation to +oo whose terms do not tend to zero, which therefore has no finite
value: sum(2^k, k, 0, +oo) and sum(n^n, n, 1, +oo) are +oo.

Remarks

The nth-term test: if the terms do not tend to zero the series diverges. That alone
says a series has no sum; it does not say what to answer instead. What settles the answer is
the sign of the limit — where the terms tend to a positive L, they are
eventually all above L / 2, so the partial sums pass every bound and the value is
+oo; a negative limit gives -oo the same way. The finitely many terms before
that point are finite and cannot change it.
A limit of zero is declined, and that is the whole difficulty of the test. It is
exactly the case the nth-term test says nothing about: sum(1/k) diverges and
sum(1/k^2) converges, and their terms both tend to 0. Answering either from this
reader would be a guess. So would a limit that does not exist — sum((-1)^k) has no
value rather than an infinite one — and that is left as written too, since telling "the
limit does not exist" apart from "the limit was not computed" is not something to infer
from a failed computation.
The summand must have no pole in the index, and this is checked before the limit is
asked for. A single undefined term makes the whole sum undefined, not infinite:
sum(k / (k - 5), k, 0, +oo) has terms tending to 1, and answering +oo would
be wrong because the term at k = 5 does not exist. Rather than hunt for poles, the
index is allowed to occur only where none can arise — under +, -, *,
and powers of the three shapes HasNoPoleInTheIndex(AngouriMath.Entity,AngouriMath.Entity.Variable,AngouriMath.Entity.Number.Integer) lists. Division by
anything containing the index, a factorial of it, a logarithm of it and the rest are
declined outright. That check is also what keeps this cheap: it runs first, so a summand
this cannot speak about never reaches the limit engine.
 Last in the chain, after every closed form, so that a series which does converge is
summed rather than tested. Asked for on the review of
#1218, where
integral(floor(x)^floor(x), x, 1, +oo) splits into sum(n^n, n, 1, +oo) and was
left as written for want of this; part of
#1212.

Members

























Angouri © 2019-2023 · Project's repo · Site's repo · Octicons · Transparency · 4378 pages online