CoCalc Public Files201-303-AH / Fichiers Sage pour les Étudiants / Exemple 4 p 19.sagewsOpen with one click!
Author: Julien Giol
Description: Fichiers Sage
Compute Environment: Ubuntu 18.04 (Deprecated)

Exploration

var('n')
n
a(n) = 2^(2*n)*3^(1-n)
a(0)
3 4
S = tuple([a(0)]) S
(3,)
for i in range (1, 56): S = S + tuple([a(i)])
show(S)
(3\displaystyle 3, 4\displaystyle 4, 163\displaystyle \frac{16}{3}, 649\displaystyle \frac{64}{9}, 25627\displaystyle \frac{256}{27}, 102481\displaystyle \frac{1024}{81}, 4096243\displaystyle \frac{4096}{243}, 16384729\displaystyle \frac{16384}{729}, 655362187\displaystyle \frac{65536}{2187}, 2621446561\displaystyle \frac{262144}{6561}, 104857619683\displaystyle \frac{1048576}{19683}, 419430459049\displaystyle \frac{4194304}{59049}, 16777216177147\displaystyle \frac{16777216}{177147}, 67108864531441\displaystyle \frac{67108864}{531441}, 2684354561594323\displaystyle \frac{268435456}{1594323}, 10737418244782969\displaystyle \frac{1073741824}{4782969}, 429496729614348907\displaystyle \frac{4294967296}{14348907}, 1717986918443046721\displaystyle \frac{17179869184}{43046721}, 68719476736129140163\displaystyle \frac{68719476736}{129140163}, 274877906944387420489\displaystyle \frac{274877906944}{387420489}, 10995116277761162261467\displaystyle \frac{1099511627776}{1162261467}, 43980465111043486784401\displaystyle \frac{4398046511104}{3486784401}, 1759218604441610460353203\displaystyle \frac{17592186044416}{10460353203}, 7036874417766431381059609\displaystyle \frac{70368744177664}{31381059609}, 28147497671065694143178827\displaystyle \frac{281474976710656}{94143178827}, 1125899906842624282429536481\displaystyle \frac{1125899906842624}{282429536481}, 4503599627370496847288609443\displaystyle \frac{4503599627370496}{847288609443}, 180143985094819842541865828329\displaystyle \frac{18014398509481984}{2541865828329}, 720575940379279367625597484987\displaystyle \frac{72057594037927936}{7625597484987}, 28823037615171174422876792454961\displaystyle \frac{288230376151711744}{22876792454961}, 115292150460684697668630377364883\displaystyle \frac{1152921504606846976}{68630377364883}, 4611686018427387904205891132094649\displaystyle \frac{4611686018427387904}{205891132094649}, 18446744073709551616617673396283947\displaystyle \frac{18446744073709551616}{617673396283947}, 737869762948382064641853020188851841\displaystyle \frac{73786976294838206464}{1853020188851841}, 2951479051793528258565559060566555523\displaystyle \frac{295147905179352825856}{5559060566555523}, 118059162071741130342416677181699666569\displaystyle \frac{1180591620717411303424}{16677181699666569}, 472236648286964521369650031545098999707\displaystyle \frac{4722366482869645213696}{50031545098999707}, 18889465931478580854784150094635296999121\displaystyle \frac{18889465931478580854784}{150094635296999121}, 75557863725914323419136450283905890997363\displaystyle \frac{75557863725914323419136}{450283905890997363}, 3022314549036572936765441350851717672992089\displaystyle \frac{302231454903657293676544}{1350851717672992089}, 12089258196146291747061764052555153018976267\displaystyle \frac{1208925819614629174706176}{4052555153018976267}, 483570327845851669882470412157665459056928801\displaystyle \frac{4835703278458516698824704}{12157665459056928801}, 1934281311383406679529881636472996377170786403\displaystyle \frac{19342813113834066795298816}{36472996377170786403}, 77371252455336267181195264109418989131512359209\displaystyle \frac{77371252455336267181195264}{109418989131512359209}, 309485009821345068724781056328256967394537077627\displaystyle \frac{309485009821345068724781056}{328256967394537077627}, 1237940039285380274899124224984770902183611232881\displaystyle \frac{1237940039285380274899124224}{984770902183611232881}, 49517601571415210995964968962954312706550833698643\displaystyle \frac{4951760157141521099596496896}{2954312706550833698643}, 198070406285660843983859875848862938119652501095929\displaystyle \frac{19807040628566084398385987584}{8862938119652501095929}, 7922816251426433759354395033626588814358957503287787\displaystyle \frac{79228162514264337593543950336}{26588814358957503287787}, 31691265005705735037417580134479766443076872509863361\displaystyle \frac{316912650057057350374175801344}{79766443076872509863361}, 1267650600228229401496703205376239299329230617529590083\displaystyle \frac{1267650600228229401496703205376}{239299329230617529590083}, 5070602400912917605986812821504717897987691852588770249\displaystyle \frac{5070602400912917605986812821504}{717897987691852588770249}, 202824096036516704239472512860162153693963075557766310747\displaystyle \frac{20282409603651670423947251286016}{2153693963075557766310747}, 811296384146066816957890051440646461081889226673298932241\displaystyle \frac{81129638414606681695789005144064}{6461081889226673298932241}, 32451855365842672678315602057625619383245667680019896796723\displaystyle \frac{324518553658426726783156020576256}{19383245667680019896796723}, 129807421463370690713262408230502458149737003040059690390169\displaystyle \frac{1298074214633706907132624082305024}{58149737003040059690390169})
show(S[25].n())
3986.48073735860\displaystyle 3986.48073735860
show(S[55].n())
2.23229593379905×107\displaystyle 2.23229593379905 \times 10^{7}

Procédure pour calculer les sommes partielles

def sommes_partielles(a, N): S = tuple([a(0)]) G = points([0, a(0)], color='darkgreen', pointsize=50) for i in range(1, N+1): S_partielle = S[i-1]+a(i) S = S + tuple([S_partielle]) G = G + points([i, S_partielle], color='darkgreen', pointsize=50) return S, G
S = sommes_partielles(a, 100)[0] G = sommes_partielles(a, 100)[1]
S[100].n()
3.74157889224863e13
G

Conclusion

On voit que la suite des sommes partielles {Sn=k=0nak}\displaystyle\left\{S_n=\sum_{k=0}^n a_k\right\} est croissante et non majorée, donc : limn+Sn=+ \lim_{n\rightarrow +\infty}S_n=+\infty et la série k=0+ak\displaystyle\sum_{k=0}^{+\infty} a_k diverge.