# Book Isle: Match summation indices and limits to expanded terms

Language: en
Canonical: [Book Isle: Match summation indices and limits to expanded terms](<https://allluckly.cn/en/guides/nasbook-summation-index-limits-expansion/>)
中文：[中文 HTML](<https://allluckly.cn/guides/nasbook-summation-index-limits-expansion/>)
English: [English HTML](<https://allluckly.cn/en/guides/nasbook-summation-index-limits-expansion/>)
Published: 2026-11-09T00:00:00+08:00
Modified: 2026-11-09T09:37:10+08:00
Author: Bison

Check term counts, parentheses and index renaming using an original one-to-four sum.

English translation of the [Chinese original](<https://allluckly.cn/guides/nasbook-summation-index-limits-expansion/>), translated on 2026-11-09. Original publication dates, attribution, code and version notes are retained.

Summation notation compresses terms into one expression; a missed lower bound or parenthesis can change the sum. Locate the author’s definition in Book Isle, expand a finite integer example manually and retain index, range and term provenance. No formula recognition, automatic derivation or proof acceptance is assumed, and a copied final value does not replace understanding each term.

## Locate the convention and complete expression

Record book, edition, chapter, example and location, preserving everything above, below and beside the summation sign. Establish permitted index values, integer increments and endpoint inclusion from the author. Do not crop away subscripts, and leave unread scan details unresolved with a locator. Create five external fields: index name, lower bound, upper bound, term expression and expansion, retaining the source position. Enlarged typography cannot supply an unseen upper bound; familiar textbook patterns do not justify inventing one.

## Define the four index values from one through four

Define the original main example with index k from one through four inclusive and term twice k plus one. List one,two,three,four: four terms because the lower bound happens to be one, not because an upper bound always counts terms. Two through four contains three terms, illustrating upper minus lower plus one for a nonempty consecutive integer range. This exercise excludes empty, infinite and nonconsecutive sums and states its convention rather than imposing one notation on every mathematics book.

## Substitute each index before adding

Substitute k equal to one,two,three,four to obtain three,five,seven,nine, totaling twenty-four. Keep each index beside its term. Evaluating only k=four gives the last term nine, not the sum; adding the indices is another error. Check that one expression generates all four values before addition. Preserve any source units and variable definitions rather than ignoring meaning because numbers appear. This example uses pure numbers, not an invented population, area or cost interpretation.

## Check whether plus one belongs inside or outside the sum

In the main expression, plus one occurs in every term, four times. If the source instead sums twice k and adds one outside the summation’s scope, two+four+six+eight is twenty, then one gives twenty-one, not twenty-four. Annotate both scopes explicitly rather than guessing from proximity. Follow an established convention where typography omits brackets. If scope remains unclear, retain the question instead of using agreement with an answer alone to decide it.

## Distinguish index renaming from changing the expression

Rename k consistently to j in an external copy, retaining limits one/four and changing the corresponding letter in the term. Expansion remains three,five,seven,nine. Changing only the index declaration leaves an undefined k. A fixed parameter n is not renamed merely because the index changes; identify varying and fixed roles. This checks index roles, not arbitrary substitution legality. Preserve both forms and consult definitions for name conflicts or nested sums, beyond this exercise’s scope.

## Relist all terms when a bound changes

Change the lower bound from one to two, retaining upper four and term twice k plus one. Expansion is five,seven,nine, totaling twenty-one: the removed term is three, not one, and there are now three terms. The earlier outside-plus-one example also totals twenty-one, but matching totals do not make their expressions equivalent. Preserve conditions, expansion and sum together, especially this accidental numerical agreement, so a final-line check cannot conceal a misread scope.

## Check placement and glyphs when typography is unclear

Use available viewing controls to inspect complete bounds rather than attaching a wrapped lower bound to a previous expression. Check a possible letter l versus digit one against definitions, retaining uncertainty if unresolved. Conversion may move subscripts or superscripts; consult a trusted original rather than assuming losslessness. Paper or an existing calculator can assist arithmetic, not establish app-executed summation. Record the expression actually used. For disagreement, check index, bounds and scope before arithmetic instead of altering the source to fit.

## Rebuild the expansion from its range after returning

Leave the reader and return to expression and definition. Without old results, list four indices and three,five,seven,nine, then twenty-four. Separately check the three-term lower-two sum and the different expansion of outside-plus-one, not merely two identical twenty-ones. Preserve all five fields, source, unread characters and personal rewrites distinctly. Reopen saved notes to confirm exported bounds remain, keeping the book separately. Completion traces every term to index and range rather than treating sigma as an automatic answer button.

## Related guides and sources

[Follow worked examples in Book Isle: track symbols, units and substitutions](<https://allluckly.cn/en/guides/nasbook-worked-example-symbol-unit-log/>); [Read induction proofs in Book Isle: align base case, hypothesis and next step](<https://allluckly.cn/en/guides/nasbook-induction-proof-hypothesis-step/>).

Checked on November 9, 2026: [Apple public version information](<https://itunes.apple.com/lookup?id=6766198898&country=cn>) lists Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. Feature scope follows the [official App Store description](<https://apps.apple.com/cn/app/id6766198898>) and the installed version. This article provides reading and checking methods you can perform; it does not claim step-by-step physical-device or private-server testing. Manual records, external tools and actually visible controls are identified separately, and personal analysis or code observations are not presented as verified automatic capabilities.

For the concept or external-tool scope, see [OpenStax: Series and Their Notations](<https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-4-series-and-their-notations>). Samples, checking sequences and observation records here are original reading exercises. External references establish no new automatic recognition, editing or inference in the app and do not replace checking the actual file or passage.