R0034-17944 · q-20261014-advance-20261104-r03-absolute-convergence-of-series-0044
Step-by-step derivation
S001 · implies
Before
source operation=absseries-conditional-alternating;original question snapshot retained;source presentation interface resolved in this page
After
EXPLICIT ordered output fields=[shift, n, terms, series_converges, absolutely_converges, certificate]
Conditions: none
Preserve the original snapshot's historical interface marker while listing the full ordered fields here,without browser JavaScript or an external lookup.Field labels are presentation metadata,not mathematical evaluation.The fourth term-zero No marks insufficiency,not a mislabeled second convergence flag.
Step identifiers and dependencies
Rule: absseries_explicit_source_interface_fields_without_javascript · Uses: G001
S002 · calculation
Before
alternating displayed term0 denominator := sum(0, 19)
After
alternating displayed term0 denominator = 19
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
alternating displayed term0 signed value := div(1, 19)
After
alternating displayed term0 signed value = 1/19
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
alternating displayed term1 denominator := sum(1, 19)
After
alternating displayed term1 denominator = 20
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
alternating displayed term1 signed value := div(-1, 20)
After
alternating displayed term1 signed value = -1/20
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
alternating displayed term2 denominator := sum(2, 19)
After
alternating displayed term2 denominator = 21
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
alternating displayed term2 signed value := div(1, 21)
After
alternating displayed term2 signed value = 1/21
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
alternating displayed term3 denominator := sum(3, 19)
After
alternating displayed term3 denominator = 22
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
alternating displayed term3 signed value := div(-1, 22)
After
alternating displayed term3 signed value = -1/22
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
alternating displayed term4 denominator := sum(4, 19)
After
alternating displayed term4 denominator = 23
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
alternating displayed term4 signed value := div(1, 23)
After
alternating displayed term4 signed value = 1/23
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
absolute series of declared alternating tail:DECLARED whole positive tail b_k=1/(k+19),k>=0;shift>0
After
b_k decreases to0:for ANY epsilon>0 choose N with N+shift>1/epsilon,then every k>=N has b_k<epsilon
Conditions: none
Increasing denominators and an Archimedean cutoff supply the all-later bound for every epsilon,not an inference from the displayed prefix.
Step identifiers and dependencies
Rule: harmonic_full_tail_terms_zero_not_observed_prefix · Uses: G001, S011
S013 · implies
Before
absolute series of declared alternating tail:same full absolute/positive series over denominators m=19,20,...
After
for every integer j>=0 large enough that2^j>=shift-1,the block m=2^j+1..2^(j+1) has2^j terms each>=1/2^(j+1);each block sum>=1/2;infinitely many disjoint blocks make partial sums unbounded
Conditions: none
Removing finitely many initial terms leaves the dyadic-block argument.Each sufficiently late block contributes at least1/2,so arbitrarily many disjoint blocks force unbounded partial sums.Terms tending to zero do not give a finite limit of partial sums.
Step identifiers and dependencies
Rule: harmonic_shifted_dyadic_blocks_diverge_not_term_limit · Uses: G001, S012
S014 · implies
Before
WHOLE declared terms=(-1)^k/(k+19),k starts0 positive;displayed first5 terms=[1/19, -1/20, 1/21, -1/22, 1/23]
After
b_k positive decreasing to0;even S_2m increases by b_2m-b_(2m+1)>=0;odd S_(2m+1) decreases;0<=S_2m<=S_(2m+1)<=b_0=1/19;gap b_2m tends0
Conditions: none
Positive decreasing magnitudes make even partial sums increase and odd partial sums decrease while bounding one another.Real completeness gives both limits,and their vanishing gap makes them equal,so the whole partial-sum sequence converges.Alternation or a term-zero condition alone is not a general criterion.
Step identifiers and dependencies
Rule: absseries_alternating_monotone_subsequences_not_alternation_alone · Uses: G001, S013
S015 · calculation
Before
alternating displayed exact partial sum := sum(1/19, -1/20, 1/21, -1/22, 1/23)
After
alternating displayed exact partial sum = 97463/2018940
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · calculation
Before
alternating next-term remainder denominator := sum(5, 19)
After
alternating next-term remainder denominator = 24
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · calculation
Before
alternating next-term absolute remainder bound := div(1, 24)
After
alternating next-term absolute remainder bound = 1/24
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · calculation
Before
alternating next signed term := mul(-1, 1/24)
After
alternating next signed term = -1/24
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S017
S019 · calculation
Before
alternating next exact partial sum := sum(97463/2018940, -1/24)
After
alternating next exact partial sum = 26681/4037880
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · implies
Before
displayed S_n=97463/2018940;adjacent S_(n+1)=26681/4037880;n=5;positive shift=19
After
common limit L lies between these two endpoints;abs(L-S_n)<=1/24;original series converges=Yes;absolute harmonic series converges=No
Conditions: none
The whole-tail alternating certificate traps the common limit between adjacent partial sums and gives the next-term error bound.The separate dyadic argument proves divergence of the absolute series.Both certificates establish conditional convergence;the displayed finite list alone proves neither infinite claim.
Step identifiers and dependencies
Rule: absseries_conditional_requires_convergence_and_absolute_divergence · Uses: G001, S019