How do you simplify 1! + 2! + 3! +...+2007!?
1 Answer
sum_(n=1)^2007 (n!) ~~ 4.3067 xx 10^5758
Explanation:
There is no factorial identity that is going to help you with this one, but you can ask a computer to calculate it for you (using Wolfram, Mathematica, etc.):
sum_(n=1)^2007 (n!) =
43,067,483,829,623,961,638,366,972,205,
981,802,752,007,314,172,910,886,370,515,
671,037,192,243,177,689,895,234,663,758,
030,046,887,820,196,823,525,909,332,425,
169,515,742,353,485,626,028,181,445,858,
838,167,390,689,873,784,469,071,595,905,
271,356,685,607,377,027,406,996,208,062,
199,376,945,065,718,903,195,361,237,502,
664,131,238,353,701,933,946,645,947,597,
436,863,715,168,075,505,697,796,445,024,
078,056,694,292,870,381,672,101,138,135,
287,669,281,334,059,277,468,209,876,144,
283,221,900,663,446,407,198,843,441,871,
996,412,882,682,937,957,248,618,348,357,
254,817,357,809,695,538,625,381,271,502,
157,631,830,102,251,253,739,721,556,947,
604,189,426,030,721,580,051,641,743,410,
653,099,348,026,410,960,458,670,597,339,
074,521,048,665,821,170,596,818,073,500,
181,073,763,329,121,135,437,458,124,802,
299,919,750,891,825,407,617,821,890,872,
802,030,694,851,391,780,979,178,782,252,
336,776,631,229,474,449,881,635,721,322,
626,361,401,031,096,542,098,210,004,735,
289,115,710,287,530,607,613,529,622,146,
867,133,387,338,891,407,878,071,624,630,
548,392,227,400,885,434,342,989,254,469,
627,664,300,295,821,808,861,760,574,813,
441,065,796,454,681,791,322,046,330,661,
163,648,779,522,030,110,552,545,799,511,
453,568,336,369,961,428,837,554,767,129,
450,200,460,762,962,773,726,684,527,342,
725,164,681,146,296,392,533,590,637,972,
521,704,191,682,120,041,210,355,904,625,
373,464,953,457,911,715,433,531,638,591,
846,482,115,563,091,190,304,578,052,931,
482,810,984,724,282,410,726,681,480,621,
032,443,636,525,128,558,426,622,190,598,
372,418,883,517,974,112,435,530,091,476,
783,905,407,578,088,337,823,159,369,549,
292,296,870,465,399,091,150,285,813,083,
800,978,643,847,470,129,044,127,991,277,
220,225,949,339,464,663,111,589,281,978,
157,433,879,922,845,386,737,932,025,193,
783,110,663,207,002,985,168,369,557,675,
700,295,977,480,092,236,320,830,557,145,
350,815,462,294,175,680,403,064,804,368,
304,946,294,125,698,816,696,210,663,629,
255,721,318,309,625,286,186,993,012,132,
150,421,985,949,546,142,775,134,166,507,
015,369,633,205,201,845,817,701,065,288,
947,040,074,802,439,903,617,532,386,258,
574,265,270,528,737,337,069,216,436,613,
367,204,727,149,666,546,289,994,379,249,
975,371,986,296,264,522,893,753,892,187,
380,364,481,714,057,286,916,071,983,739,
648,447,913,547,145,636,536,430,669,688,
022,330,659,945,325,260,019,241,093,371,
567,311,598,381,742,040,426,748,338,767,
496,821,753,748,456,396,523,410,928,733,
626,948,435,340,841,294,061,270,778,873,
179,775,443,505,173,502,512,011,563,511,
771,228,435,865,415,819,025,245,843,310,
990,530,075,324,867,669,886,756,433,699,
296,399,487,095,239,845,507,519,176,495,
326,556,049,375,911,875,465,788,378,627,
411,072,614,662,518,923,389,374,259,028,
134,090,843,489,524,241,492,012,190,093,
088,703,220,222,668,218,572,181,653,958,
688,131,963,105,300,508,083,202,502,419,
504,631,652,051,404,042,103,199,278,985,
409,429,660,323,039,929,561,775,991,291,
560,136,576,527,461,455,356,428,548,478,
297,322,235,986,770,610,210,519,588,910,
928,017,202,311,596,013,180,253,107,055,
009,471,919,967,743,791,412,740,617,714,
180,944,295,946,805,135,430,451,745,671,
855,964,298,110,133,425,281,174,172,862,
486,100,837,779,710,507,294,053,969,606,
467,939,433,473,569,154,987,696,962,638,
225,463,149,504,382,550,471,839,372,898,
418,866,302,803,120,240,979,390,834,524,
860,542,741,200,576,316,764,624,861,722,
308,434,333,104,582,642,760,955,926,339,
150,838,186,906,527,491,727,006,671,380,
616,397,242,510,381,497,585,166,865,187,
394,057,507,632,047,084,732,626,448,365,
967,574,256,333,031,613,559,229,911,347,
243,990,775,447,342,469,229,120,871,228,
818,014,773,345,462,625,159,576,789,247,
744,898,854,122,858,219,716,088,712,617,
550,051,695,465,802,145,186,870,137,976,
781,984,989,215,132,017,912,589,347,233,
770,956,711,578,592,531,881,612,354,869,
407,359,278,193,086,953,719,823,387,292,
735,855,914,779,675,675,671,446,615,287,
015,450,821,641,359,933,376,992,676,118,
055,101,101,051,110,644,286,961,500,489,
297,203,700,805,785,521,223,014,711,159,
355,860,776,113,389,829,302,561,350,891,
651,128,384,992,751,546,940,663,031,720,
892,399,009,172,689,079,907,026,895,205,
751,834,581,078,379,752,718,331,773,726,
490,320,179,632,680,993,000,028,574,028,
458,057,669,968,742,208,102,974,101,072,
453,500,723,389,444,125,730,323,893,226,
941,177,969,916,885,381,111,178,587,812,
075,564,286,080,362,431,192,490,922,418,
592,711,933,835,862,323,938,393,479,065,
264,265,726,277,509,905,610,631,737,776,
445,577,774,824,223,858,477,481,497,646,
799,767,699,186,705,813,142,837,487,946,
974,635,397,089,221,484,899,932,659,508,
885,249,876,439,053,840,031,240,374,448,
335,690,878,575,682,713,905,464,065,688,
143,406,406,862,676,583,529,400,969,563,
322,577,466,222,415,830,840,983,444,853,
596,462,892,469,995,262,235,420,313,850,
827,950,409,474,023,389,305,391,030,729,
495,202,285,164,841,449,153,204,497,691,
871,943,342,079,108,024,172,330,280,389,
115,177,543,826,363,628,391,337,101,247,
901,972,233,052,112,250,028,222,369,235,
716,753,246,717,918,081,400,310,457,556,
852,728,800,444,426,609,133,194,559,493,
428,971,200,501,910,447,244,200,554,026,
922,191,587,129,824,233,287,537,275,027,
358,742,399,326,272,879,949,083,551,397,
628,959,294,382,536,785,108,698,075,354,
221,096,092,545,841,484,188,090,706,544,
120,700,973,462,086,838,205,874,378,520,
533,109,071,026,973,971,748,502,430,338,
135,147,904,822,893,610,348,818,037,833,
413,691,629,868,910,014,946,844,649,891,
351,143,623,222,859,092,432,086,076,251,
166,934,397,753,607,142,303,123,697,449,
500,478,164,059,694,670,284,189,842,637,
840,169,303,089,677,512,608,701,464,344,
613,565,925,010,314,725,694,446,370,113,
831,902,590,863,825,118,442,697,989,263,
085,428,088,406,989,871,127,187,279,450,
511,964,198,885,459,422,148,457,349,833,
436,493,056,297,307,912,230,446,810,542,
885,102,446,711,901,076,520,544,383,446,
608,263,761,527,151,275,937,263,141,242,
760,304,123,665,782,021,121,307,019,097,
949,041,572,274,284,773,408,899,586,713,
728,673,915,273,383,601,896,681,723,810,
102,749,597,995,862,024,317,987,827,375,
296,928,597,508,460,174,434,508,108,073,
723,238,194,219,605,672,980,551,680,424,
552,141,747,249,263,325,753,066,503,808,
627,717,352,539,916,020,133,217,662,920,
319,027,359,129,388,144,082,049,119,701,
752,987,471,960,292,930,696,708,810,755,
066,804,286,133,510,338,965,270,517,156,
018,430,793,724,601,562,445,998,153,373,
821,130,997,172,979,460,850,584,648,207,
952,544,407,979,517,701,139,501,230,325,
170,685,307,140,382,354,643,613,128,281,
828,047,147,158,192,682,251,704,970,640,
225,499,853,860,304,437,564,868,257,304,
873,686,798,610,292,152,371,976,753,152,
466,896,044,486,439,515,528,443,015,985,
707,863,240,922,411,392,699,825,134,668,
799,287,207,921,670,525,667,465,483,896,
047,500,760,845,957,282,616,908,787,514,
362,961,677,541,700,318,751,278,644,034,
929,174,650,363,511,672,265,373,044,378,
450,515,108,772,008,339,801,176,044,350,
993,239,096,973,028,570,280,468,894,058,
859,859,112,987,286,488,125,028,062,633,
884,400,732,946,635,275,544,887,024,740,
984,199,258,212,525,241,456,933,829,801,
981,088,109,375,463,882,399,728,280,761,
109,622,101,995,746,925,171,844,037,841,
823,522,441,611,867,345,433,208,379,741,
347,009,406,659,919,379,208,878,761,799,
827,677,548,449,306,681,397,552,694,707,
427,117,631,406,144,007,450,882,969,740,
049,818,323,455,858,551,317,207,197,201,
589,704,858,224,388,374,914,095,841,726,
938,767,482,243,085,025,897,244,690,463,
595,978,336,569,707,778,625,455,613,570,
273,121,541,638,995,145,894,561,079,999,
213,500,076,101,248,781,500,642,979,091,
198,044,736,325,729,107,144,579,768,273,
776,743,716,439,557,581,557,253,878,204,
647,391,176,582,851,167,312,182,026,188,
795,156,620,056,856,503,340,092,247,479,
478,684,738,621,107,994,804,323,593,105,
039,052,556,442,336,528,920,420,940,313
