AngouriMath

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AngouriMath

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GeometricSeries

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Description

Summary

A summation whose summand is a power with the index in the exponent, times something free
of the index: sum(x^k, k, 0, n) is (1 - x^(n + 1)) / (1 - x), and
sum(2^(-k), k, 0, +oo) is 2.

Remarks

The summand is read as C * b^(m k + s), which is C b^s * r^k with the ratio
r = b^m for a whole non-zero m. Between two bounds the sum is
C b^s (r^a - r^(b + 1)) / (1 - r), which holds for every pair of integers with
b >= a - 1 and every ratio but 1 -- at r = 1 the sum is the number of terms
times the constant, and below b = a - 1 the range is empty, which this library
answers with 0. Where the ratio is a number the branch that applies is known; where it
is symbolic, both are offered as a piecewise, the way
PolynomialSummation offers the empty range.
 To +oo the series converges exactly when |r| < 1, to C b^s r^a / (1 - r).
A numeric ratio outside that is left as written -- the sum is infinite or has no value,
and which of the two is not this reader's to say -- and a symbolic ratio carries the
condition, since outside it the sum has no value the formula could be standing for.
 Recognised structurally, on the factors of the simplified summand: exactly one power whose
base is free of the index and whose exponent is the index times a whole number plus
something free of the index; every other factor free of the index. A polynomial factor in
the index -- k x^k -- is not this family and is declined, and so is a base that is
zero or a ratio that is a number and 1. Part of question I.2 of
#1212, where the
step integral of 2^(-floor(x)) leaves this sum.

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