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Svmmationes, progressiones et indvctio

Svmmarivm

Haec lectio notationem svmmationis eivsqve proprietates (linearitatem, decompositionem, translationem indicis et inversionem ordinis) introdvcit, eamqve ad svmmas telescopicas et ad svmmas progressionvm arithmeticarvm et geometricarvm compvtandas adhibet. Deinde indvctionem fortem et principivm bonae ordinationis tractat, ostendit indvctionem vtrvmqve implicare, definit qvo sensv ei aeqvivalentia sint, eaqve adhibet ad ostendendvm omnem nvmervm natvralem maiorem qvam 11 prodvctvm nvmerorvm primorvm esse.

Proposita discendi

  1. Svmmas notatione svmmationis scribere et transformare, eivs proprietates linearitatis, decompositionis, translationis indicis et inversionis ordinis adhibendo.
  2. Svmmas telescopicas compvtare easqve adhibere ad formvlas svmmarvm dedvcendas, non solvm confirmandas.
  3. Terminvm generalem et svmmam primorvm terminorvm progressionis arithmeticae vel geometricae determinare, eaqve ad problemata solvenda adhibere.
  4. Per indvctionem fortem et per principivm bonae ordinationis demonstrare, vtrvmqve ex principio indvctionis dedvcere et exponere qvibvs factis additis bona ordinatio indvctionem restitvere sinat.

Notatio svmmationis

In lectione de axiomatibvs Peani svmmae mvltorvm terminorvm apparvervnt qvi regvlam seqvvntvr, vt 1+2+⋯+n1 + 2 + \cdots + n. Pvncta svspensionis lectori mvnvs eivs regvlae divinandae committvnt; notatio svmmationis eam declarat. Si am,am+1,…,ana_m, a_{m + 1}, \ldots, a_n nvmeri svnt, cvm m≤nm \leq n, scribitvr

∑k=mnak=am+am+1+⋯+an\sum_{k=m}^{n} a_k = a_m + a_{m + 1} + \cdots + a_n

et legitvr «svmma terminorvm aka_k pro kk a mm ad nn». Littera kk est index svmmae; mm et nn svnt eivs termini inferior et svperior, et aka_k est terminvs generalis. Svmma n−m+1n - m + 1 terminos habet. Index est variabilis mvta: valor svmmae non pendet a littera qva nominatvr, ita vt ∑k=1nk2\sum_{k=1}^{n} k^2 et ∑j=1nj2\sum_{j=1}^{n} j^2 idem nvmervs sint. Index nvmeros natvrales percvrrit; termini avtem nvmeri alivs generis esse possvnt. In hac lectione apparent, praeter nvmeros natvrales, nvmeri integri cvm svis negativis et fractiones, qvos pars seqvens constrvit: hic anticipantvr, et eorvm solae regvlae elementares calcvli et ordinis adhibentvr (commvtativitas, associativitas, distribvtivitas, regvlae signorvm, svbtractio et divisio per nvmervm non nvllvm). Proprietates svmmationis qvae seqvvntvr his regvlis demonstrantvr, ideoqve pro terminis cvivsvis horvm genervm valent.

Sicvt additio, mvltiplicatio et potentia in lectione de operationibvs nvmerorvm natvralivm, svmmatio definitionem recvrsivam admittit qvae pvnctis svspensionis caret: svmma vnivs termini est ipse terminvs, et vt terminvs addatvr, ad svmmam priorem additvr.

∑k=mmak=am,∑k=mn+1ak=∑k=mnak+an+1\sum_{k=m}^{m} a_k = a_m, \qquad \sum_{k=m}^{n + 1} a_k = \sum_{k=m}^{n} a_k + a_{n + 1}

Haec definitio a mm incipit, cvm theorema recvrsionis, vt envntiatvm est, fvnctiones a 11 definiat. Satis tamen est indicem transferre: pro svmma vsqve ad nn definitvr svmma qq primorvm terminorvm a ama_m incipiens. Cvm valore initiali ama_m et regvla gq(x)=x+am+qg_q(x) = x + a_{m + q}, theorema recvrsionis vnicam fvnctionem FF ex N\mathbb{N} in nvmeros praebet talem vt F(1)=amF(1) = a_m et F(q+1)=F(q)+am+qF(q + 1) = F(q) + a_{m + q} pro omni nvmero natvrali qq. Pro n≥mn \geq m svmma vsqve ad nn tvm definitvr vt valor fvnctionis FF in nvmero terminorvm eivs, n−m+1n - m + 1, qvi nvmervs natvralis est qvia m<n+1m < n + 1:

∑k=mnak=F(n−m+1)\sum_{k=m}^{n} a_k = F(n - m + 1)

Cvm n=mn = m obtinetvr F(1)=amF(1) = a_m, et cvm q=n−m+1q = n - m + 1, qvia m+q=n+1m + q = n + 1, aeqvalitas F(q+1)=F(q)+am+qF(q + 1) = F(q) + a_{m + q} est ipsa secvnda regvla definitionis. Vnde definitio recvrsiva a mm incipiens vnicam svmmam pro qvoqve n≥mn \geq m determinat, et si m=1m = 1, congrvit cvm ea qvam theorema recvrsionis directe praebet, qvia tvm F(n)=∑k=1nakF(n) = \sum_{k=1}^{n} a_k.

Hac notatione, formvla per indvctionem demonstrata in lectione de axiomatibvs Peani, scilicet svmmae nn primorvm nvmerorvm natvralivm, sic scribitvr:

∑k=1nk=n(n+1)2\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}

Svmma qvadratorvm, ∑k=1nk2\sum_{k=1}^{n} k^2, proxime seqvitvr, et haec lectio non solvm eam confirmabit: sectio de svmmis telescopicis eam dedvcit non privs cognitam.

Indvctiones in terminvm svperiorem svmmae a termino inferiore mm incipivnt, qvi non necessario est 11. Principivm tamen indvctionis axiomatvm Peani a 11 incipit; vt ab alio nvmero natvrali incipiatvr, lemmate seqventi opvs est.

LemmaIndvctio a qvolibet nvmero natvrali

Sit n0n_0 nvmervs natvralis. Si proprietas pro n0n_0 valet et, posito eam pro qvolibet nvmero natvrali k≥n0k \geq n_0 valere, etiam pro k+1k + 1 valet, tvm ea proprietas pro omnibvs nvmeris natvralibvs n≥n0n \geq n_0 valet. Consilivm est nvmeros natvrales a n0n_0 denvo nvmerare, vt ad principivm indvctionis ordinarivm redeatvr. Adhibentvr lemma lectionis de operationibvs nvmerorvm natvralivm, secvndvm qvod omnis nvmervs natvralis a 11 diversvs svccessor est, et monotonia additionis. In figvra, n0=4n_0 = 4 et p=3p = 3: svb qvoqve nvmero natvrali a n0n_0 novvs eivs nvmervs qq apparet.

Demonstratio

  1. n0=p+1,Q(q)↔P(p+q)n_0 = p + 1, \quad Q(q) \leftrightarrow P(p + q)

    Si n0=1n_0 = 1, est ipsvm principivm indvctionis. Sin minvs, n0=p+1n_0 = p + 1 pro aliqvo nvmero natvrali pp, et proprietatem QQ consideramvs qvae PP pro p+qp + q affirmat.

  2. Q(1)↔P(p+1)↔P(n0)Q(1) \leftrightarrow P(\resaltar{p + 1}) \leftrightarrow P(\resaltar{n_0})

    Proprietas QQ pro 11 valet, qvia Q(1)Q(1) est P(n0)P(n_0).

  3. P(p+q)→P((p+q)+1)=P(p+(q+1))P(p + q) \rightarrow P(\resaltar{(p + q) + 1}) = P(\resaltar{p + (q + 1)})

    Si Q(q)Q(q) valet, qvia 1≤q1 \leq q, per monotoniam additionis est p+q≥p+1=n0p + q \geq p + 1 = n_0, et gradvs datvs ad k=p+qk = p + q adhibetvr: Q(q+1)Q(q + 1) valet.

  4. n≥n0→(∃q∈N)(n=p+q)n \geq n_0 \rightarrow (\exists q \in \mathbb{N})(n = p + q)

    Per principivm indvctionis, Q(q)Q(q) pro omni qq valet. Omnis n≥n0n \geq n_0 formam p+qp + q habet: si n=n0n = n_0, cvm q=1q = 1, et si n=n0+ln = n_0 + l, cvm q=1+lq = 1 + l.

Proprietates svmmationis

Ex definitione recvrsiva, per indvctionem in terminvm svperiorem, proprietates seqventes dedvcvntvr, qvae pro terminis qvibvslibet aka_k et bkb_k et pro qvolibet nvmero cc valent:

  1. Linearitas: ∑k=mn(ak+bk)=∑k=mnak+∑k=mnbk\sum_{k=m}^{n} (a_k + b_k) = \sum_{k=m}^{n} a_k + \sum_{k=m}^{n} b_k et ∑k=mnc ak=c∑k=mnak\sum_{k=m}^{n} c\,a_k = c \sum_{k=m}^{n} a_k. Praesertim, constantem nn vicibvs addere idem est atqve eam per nn mvltiplicare: ∑k=1nc=nc\sum_{k=1}^{n} c = nc.
  2. Decompositio: si m≤p<nm \leq p < n, tvm ∑k=mnak=∑k=mpak+∑k=p+1nak\sum_{k=m}^{n} a_k = \sum_{k=m}^{p} a_k + \sum_{k=p+1}^{n} a_k.
  3. Translatio indicis: pro omni nvmero natvrali pp, ∑k=mnak=∑j=m+pn+paj−p\sum_{k=m}^{n} a_k = \sum_{j=m+p}^{n+p} a_{j - p}.
  4. Inversio ordinis: ∑k=1nak=∑j=1nan+1−j\sum_{k=1}^{n} a_k = \sum_{j=1}^{n} a_{n + 1 - j}.
TheoremaProprietates svmmationis

Pro terminis qvibvslibet aka_k et bkb_k et pro qvolibet nvmero cc, svmmatio linearis est, in dvas svmmas consecvtivas decomponitvr, translationem indicis et inversionem ordinis admittit, sicvt in elencho svperiore dictvm est. Qvaeqve proprietas per indvctionem in terminvm svperiorem demonstratvr, qvae a termino inferiore incipit beneficio lemmatis svperioris. Solae adhibentvr definitio recvrsiva et proprietates additionis et mvltiplicationis; inversio praeterea decompositione et translatione nititvr.

Demonstratio

  1. ∑k=mn+1(ak+bk)=∑k=mnak+∑k=mnbk+(an+1+bn+1)=∑k=mn+1ak+∑k=mn+1bk\sum_{k=m}^{n + 1} (a_k + b_k) = \resaltar{\sum_{k=m}^{n} a_k + \sum_{k=m}^{n} b_k} + (a_{n + 1} + b_{n + 1}) = \sum_{k=m}^{n + 1} a_k + \sum_{k=m}^{n + 1} b_k

    Linearitas. Pro n=mn = m, ambo latera am+bma_m + b_m valent. Si pro nn valet, adhibentvr definitio recvrsiva, hypothesis indvctionis et associativitas et commvtativitas additionis. Eodem modo, distribvtivitate adhibita, ostenditvr ∑k=mnc ak=c∑k=mnak\sum_{k=m}^{n} c\,a_k = c \sum_{k=m}^{n} a_k, et ∑k=1nc=nc\sum_{k=1}^{n} c = nc, qvia nc+c=(n+1)cnc + c = (n + 1)c.

  2. ∑k=mp+1ak=∑k=mpak+ap+1=∑k=mpak+∑k=p+1p+1ak\sum_{k=m}^{p + 1} a_k = \sum_{k=m}^{p} a_k + \resaltar{a_{p + 1}} = \sum_{k=m}^{p} a_k + \resaltar{\sum_{k=p+1}^{p + 1} a_k}

    Decompositio, pp fixo, per indvctionem in nn a p+1p + 1 incipientem. Pro n=p+1n = p + 1 est ipsa definitio recvrsiva, qvia svmma vnivs termini est ipse terminvs.

  3. ∑k=mn+1ak=∑k=mpak+∑k=p+1nak+an+1=∑k=mpak+∑k=p+1n+1ak\sum_{k=m}^{n + 1} a_k = \resaltar{\sum_{k=m}^{p} a_k + \sum_{k=p+1}^{n} a_k} + a_{n + 1} = \dato{d}{\sum_{k=m}^{p} a_k + \sum_{k=p+1}^{n + 1} a_k}

    Si decompositio pro nn valet, terminvs an+1a_{n + 1} vtriqve lateri additvr et definitio recvrsiva ad qvamqve svmmam adhibetvr.

  4. ∑j=m+pn+1+paj−p=∑j=m+pn+paj−p+an+1=∑k=mnak+an+1=∑k=mn+1ak\sum_{j=m+p}^{n+1+p} a_{j - p} = \resaltar{\sum_{j=m+p}^{n+p} a_{j - p}} + a_{n + 1} = \resaltar{\sum_{k=m}^{n} a_k} + a_{n + 1} = \dato{s}{\sum_{k=m}^{n + 1} a_k}

    Translatio. Pro n=mn = m, ambo latera ama_m valent. Si pro nn valet, qvia terminvs svperior (n+p)+1(n + p) + 1 est (n+1)+p(n + 1) + p, adhibentvr definitio recvrsiva et hypothesis indvctionis.

  5. ∑j=1n+1an+2−j=an+1+∑j=2n+1an+2−j=an+1+∑i=1nan+1−i\sum_{j=1}^{n + 1} a_{n + 2 - j} = a_{n + 1} + \resaltar{\sum_{j=2}^{n + 1} a_{n + 2 - j}} = a_{n + 1} + \resaltar{\sum_{i=1}^{n} a_{n + 1 - i}}

    Inversio, per indvctionem in nn. Pro n=1n = 1, ambo latera a1a_1 valent. Si pro nn valet, primvs terminvs svmmae inversae n+1n + 1 terminorvm per decompositionem separatvr, et index vno transfertvr, cvm j=i+1j = i + 1.

  6. an+1+∑i=1nan+1−i=an+1+∑k=1nak=∑k=1n+1aka_{n + 1} + \resaltar{\sum_{i=1}^{n} a_{n + 1 - i}} = a_{n + 1} + \resaltar{\sum_{k=1}^{n} a_k} = \sum_{k=1}^{n + 1} a_k

    Per hypothesim indvctionis, svmma vltima ∑k=1nak\sum_{k=1}^{n} a_k valet; per commvtativitatem et definitionem recvrsivam, inversio pro n+1n + 1 obtinetvr.

Linearitas sinit svmmam implicatam ad svmmas notas redvcere. Dvae vltimae proprietates svnt mvtationes variabilis: translatio k=j−pk = j - p svbstitvit et eosdem terminos alia nvmeratione percvrrit, et inversio k=n+1−jk = n + 1 - j svbstitvit eosqve ab vltimo ad primvm percvrrit. Svmmatio tamen non eodem modo cvm prodvcto se gerit: svmma prodvctorvm non est prodvctvm svmmarvm. Satis est exemplvm cvm dvobvs terminis et ak=bk=ka_k = b_k = k:

∑k=12k⋅k=1+4=5≠9=(1+2)(1+2)\sum_{k=1}^{2} k \cdot k = 1 + 4 = 5 \neq 9 = (1 + 2)(1 + 2)
PropositioSvmma primorvm nvmerorvm imparivm

Demonstrare svmmam nn primorvm nvmerorvm imparivm esse n2n^2. Linearitas svmmam ad dvas alias iam notas redvcit, nvlla nova indvctione opvs.

Demonstratio

  1. 1+3+5+⋯+(2n−1)=∑k=1n(2k−1)1 + 3 + 5 + \cdots + (2n - 1) = \dato{s}{\sum_{k=1}^{n} (2k - 1)}

    Impar qvi locvm kk obtinet est 2k−12k - 1: primvs est 11, secvndvs 33 et is qvi locvm nn obtinet est 2n−12n - 1. Svmma qvaesita sic scribitvr.

  2. ∑k=1n(2k−1)=2∑k=1nk−∑k=1n1\sum_{k=1}^{n} \resaltar{(2k - 1)} = \resaltar{2 \sum_{k=1}^{n} k - \sum_{k=1}^{n} 1}

    Per linearitatem, svmma in dvas separatvr: dvplvm svmmae nvmerorvm kk, minvs svmma nn vnitatvm.

  3. 2∑k=1nk−∑k=1n1=n(n+1)−n2 \resaltar{\sum_{k=1}^{n} k} - \resaltar{\sum_{k=1}^{n} 1} = \resaltar{n(n + 1)} - \resaltar{n}

    Prima svmma est svmma nn primorvm nvmerorvm natvralivm, n(n+1)2\frac{n(n + 1)}{2}, et factor 22 eivs denominatorem tollit; secvnda nn vicibvs nvmervm 11 addit.

  4. n(n+1)−n=n2+n−n=n2\resaltar{n(n + 1)} - n = \resaltar{n^2 + n} - n = n^2

    Prodvctvm evolvimvs; termini nn et −n-n se invicem tollvnt.

  5. k2−(k−1)2=2k−1k^2 - (k - 1)^2 = 2k - 1

    Figvra cavsam ostendit: qvisqve impar est differentia dvorvm qvadratorvm consecvtivorvm, gnomon qvi qvadratvm lateris k−1k - 1 in qvadratvm lateris kk convertit.