feat(Analysis): integral test for infinite sums

[ajirving]

Extends the results in SumIntegralComparisons.lean to infinite sums. If a function is nonnegative, antitone and integrable on $(0,
\infty)$ then we prove that its restriction to $\N$ is summable and we
bound its sum by the integral.

This upstreams some results I recently added to the PrimeNumberTheoremAnd project in Mertens.lean.

---

[github-actions]

PR summary 5758a7511d

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ AntitoneOn.sum_range_le_integral
+ AntitoneOn.summable_of_integrable
+ AntitoneOn.tsum_add_one_le_integral
+ AntitoneOn.tsum_le_integral

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 5758a75).

• +4 new declarations
• −0 removed declarations

+AntitoneOn.sum_range_le_integral
+AntitoneOn.summable_of_integrable
+AntitoneOn.tsum_add_one_le_integral
+AntitoneOn.tsum_le_integral

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 5758a7511d
Reference commit 0c9776382e

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).