1-1/2-1/3-1/4-……-1/n的和求公式
2022-12-21
1-1/2-1/3-1/4-……-1/n
= 2-1-1/2-1/3-1/4-……-1/n
= 2-(1/1+1/2+1/3+1/4+......+1/n)
括号中为正整数倒数,也称作调和数列。迄今为止,数学家们还没有找到调和数列的求和公式。
因此,本题所列算式,没有直接的求和公式。只能手工计算,或者编程,或者利用在线工具求解。
编程计算了n=[2,100]的近似值,如下:
S2 = 0.50000000000000;S3 = 0.16666666666667;S4 = -0.08333333333333;
S5 = -0.28333333333333;S6 = -0.45000000000000;S7 = -0.59285714285714;
S8 = -0.71785714285714;S9 = -0.82896825396825;S10 = -0.92896825396825;
S11 = -1.01987734487734;S12 = -1.10321067821068;S13 = -1.18013375513375;
S14 = -1.25156232656233;S15 = -1.31822899322899;S16 = -1.38072899322899;
S17 = -1.43955252264076;S18 = -1.49510807819631;S19 = -1.54773965714368;
S20 = -1.59773965714368;S21 = -1.64535870476273;S22 = -1.69081325021727;
S23 = -1.73429151108684;S24 = -1.77595817775351;S25 = -1.81595817775351;
S26 = -1.85441971621505;S27 = -1.89145675325208;S28 = -1.92717103896637;
S29 = -1.96165379758706;S30 = -1.99498713092039;S31 = -2.02724519543652;
S32 = -2.05849519543652;S33 = -2.08879822573955;S34 = -2.11820999044543;
S35 = -2.14678141901686;S36 = -2.17455919679464;S37 = -2.20158622382167;
S38 = -2.22790201329535;S39 = -2.25354303893638;S40 = -2.27854303893638;
S41 = -2.30293328283881;S42 = -2.32674280664834;S43 = -2.34999862060183;
S44 = -2.37272589332910;S45 = -2.39494811555132;S46 = -2.41668724598610;
S47 = -2.43796384173079;S48 = -2.45879717506412;S49 = -2.47920533832943;
S50 = -2.49920533832943;S51 = -2.51881318146668;S52 = -2.53804395069745;
S53 = -2.55691187522575;S54 = -2.57543039374427;S55 = -2.59361221192609;
S56 = -2.61146935478323;S57 = -2.62901321443235;S58 = -2.64625459374270;
S59 = -2.66320374628507;S60 = -2.67987041295174;S61 = -2.69626385557469;
S62 = -2.71239288783275;S63 = -2.72826590370577;S64 = -2.74389090370577;
S65 = -2.75927551909038;S66 = -2.77442703424190;S67 = -2.78935240737623;
S68 = -2.80405828972917;S69 = -2.81855104335236;S70 = -2.83283675763807;
S71 = -2.84692126468032;S72 = -2.86081015356921;S73 = -2.87450878370620;
S74 = -2.88802229721971;S75 = -2.90135563055305;S76 = -2.91451352528989;
S77 = -2.92750053827690;S78 = -2.94032105109741;S79 = -2.95297927894552;
S80 = -2.96547927894552;S81 = -2.97782495795786;S82 = -2.99002007990908;
S83 = -3.00206827268017;S84 = -3.01397303458493;S85 = -3.02573774046728;
S86 = -3.03736564744402;S87 = -3.04885990031759;S88 = -3.06022353668122;
S89 = -3.07145949173740;S90 = -3.08257060284852;S91 = -3.09355961383753;
S92 = -3.10442917905492;S93 = -3.11518186722696;S94 = -3.12582016509930;
S95 = -3.13634648088877;S96 = -3.14676314755544;S97 = -3.15707242590596;
S98 = -3.16727650753861;S99 = -3.17737751763962;S100 = -3.18737751763962;
附:fortran代码,供参考