Home » Archimedes archive » Acorn User » AU 1997-01 B.adf » Regulars » StarInfo/Allen/!Ignotum/Formulae/Formulae/Seri
StarInfo/Allen/!Ignotum/Formulae/Formulae/Seri
This website contains an archive of files for the Acorn Electron, BBC Micro, Acorn Archimedes, Commodore 16 and Commodore 64 computers, which Dominic Ford has rescued from his private collection of floppy disks and cassettes.
Some of these files were originally commercial releases in the 1980s and 1990s, but they are now widely available online. I assume that copyright over them is no longer being asserted. If you own the copyright and would like files to be removed, please contact me.
| Tape/disk: | Home » Archimedes archive » Acorn User » AU 1997-01 B.adf » Regulars |
| Filename: | StarInfo/Allen/!Ignotum/Formulae/Formulae/Seri |
| Read OK: | ✔ |
| File size: | 05FF bytes |
| Load address: | 0000 |
| Exec address: | 0000 |
File contents
# Maths > Series # New entries should take the form of: # Formula # Note 1 # Note 2 # Note 3 # Note 4 # Note 5 # Formula # Note 1 # And so on... # To fit snugly into the window, each line should be no longer # than 42 characters. # There is a limit of 25 formulas per topic. # Any notes should be made here, at the beginning and should # be preceeded by a hash (#). nth term=a+(n-1)d For an arithmetic progression, eg 1 3 5 7 9 etc. a=starting number (eg 1) d=common difference (eg 2) S<n>=(n/2)(2a+(n-1)d) For an arithmetic progression, eg 1 3 5 7 9 etc. S<n>=sum of the numbers up to term n. a=starting number (eg 1) d=common difference (eg 2) nth term=ar^(n-1) For a geometric progression, eg 1 2 4 8 16 a=starting number (eg 1) r=common ratio (eg 2) S<n>=a(1-r^n)/(1-r) For a geometric progression, eg 1 2 4 8 16 S<n>=sum of the numbers up to term n. a=starting number (eg 1) r=common ratio (eg 2) S<i>=a/(1-r) For a geometric progression, when |r|<1, eg 1 1/2 1/4 1/8 1/16 S<i>=sum of terms to infinity a=starting number (eg 1) r=common ratio (eg �) e=(1/0!)+(1/1!)+(1/2!)+(1/3!)+(1/4!)... This is one way of calculating e (2.718), where a! is the factorial of a: a(a-1)(a-2)(a-3).. until a-n=1 Note: 0!=1 (1+x)^n=1+[nx]+[n(n-1)x�/2!]+... Known as the power series, for expansion of the bracket (1+x)^n where n is any real number (positive, negative, integer and non-integer).
00000000 23 20 4d 61 74 68 73 20 3e 20 53 65 72 69 65 73 |# Maths > Series| 00000010 0a 0a 23 20 4e 65 77 20 65 6e 74 72 69 65 73 20 |..# New entries | 00000020 73 68 6f 75 6c 64 20 74 61 6b 65 20 74 68 65 20 |should take the | 00000030 66 6f 72 6d 20 6f 66 3a 0a 23 20 20 20 20 20 46 |form of:.# F| 00000040 6f 72 6d 75 6c 61 0a 23 20 20 20 20 20 4e 6f 74 |ormula.# Not| 00000050 65 20 31 0a 23 20 20 20 20 20 4e 6f 74 65 20 32 |e 1.# Note 2| 00000060 0a 23 20 20 20 20 20 4e 6f 74 65 20 33 0a 23 20 |.# Note 3.# | 00000070 20 20 20 20 4e 6f 74 65 20 34 0a 23 20 20 20 20 | Note 4.# | 00000080 20 4e 6f 74 65 20 35 0a 23 20 20 20 20 20 46 6f | Note 5.# Fo| 00000090 72 6d 75 6c 61 0a 23 20 20 20 20 20 4e 6f 74 65 |rmula.# Note| 000000a0 20 31 0a 23 20 20 20 20 20 41 6e 64 20 73 6f 20 | 1.# And so | 000000b0 6f 6e 2e 2e 2e 0a 23 20 54 6f 20 66 69 74 20 73 |on....# To fit s| 000000c0 6e 75 67 6c 79 20 69 6e 74 6f 20 74 68 65 20 77 |nugly into the w| 000000d0 69 6e 64 6f 77 2c 20 65 61 63 68 20 6c 69 6e 65 |indow, each line| 000000e0 20 73 68 6f 75 6c 64 20 62 65 20 6e 6f 20 6c 6f | should be no lo| 000000f0 6e 67 65 72 0a 23 20 74 68 61 6e 20 34 32 20 63 |nger.# than 42 c| 00000100 68 61 72 61 63 74 65 72 73 2e 0a 23 20 54 68 65 |haracters..# The| 00000110 72 65 20 69 73 20 61 20 6c 69 6d 69 74 20 6f 66 |re is a limit of| 00000120 20 32 35 20 66 6f 72 6d 75 6c 61 73 20 70 65 72 | 25 formulas per| 00000130 20 74 6f 70 69 63 2e 0a 0a 23 20 41 6e 79 20 6e | topic...# Any n| 00000140 6f 74 65 73 20 73 68 6f 75 6c 64 20 62 65 20 6d |otes should be m| 00000150 61 64 65 20 68 65 72 65 2c 20 61 74 20 74 68 65 |ade here, at the| 00000160 20 62 65 67 69 6e 6e 69 6e 67 20 61 6e 64 20 73 | beginning and s| 00000170 68 6f 75 6c 64 0a 23 20 62 65 20 70 72 65 63 65 |hould.# be prece| 00000180 65 64 65 64 20 62 79 20 61 20 68 61 73 68 20 28 |eded by a hash (| 00000190 23 29 2e 0a 0a 6e 74 68 20 74 65 72 6d 3d 61 2b |#)...nth term=a+| 000001a0 28 6e 2d 31 29 64 0a 46 6f 72 20 61 6e 20 61 72 |(n-1)d.For an ar| 000001b0 69 74 68 6d 65 74 69 63 20 70 72 6f 67 72 65 73 |ithmetic progres| 000001c0 73 69 6f 6e 2c 20 65 67 20 31 20 33 20 35 20 37 |sion, eg 1 3 5 7| 000001d0 0a 39 20 65 74 63 2e 0a 61 3d 73 74 61 72 74 69 |.9 etc..a=starti| 000001e0 6e 67 20 6e 75 6d 62 65 72 20 28 65 67 20 31 29 |ng number (eg 1)| 000001f0 0a 64 3d 63 6f 6d 6d 6f 6e 20 64 69 66 66 65 72 |.d=common differ| 00000200 65 6e 63 65 20 28 65 67 20 32 29 0a 0a 53 3c 6e |ence (eg 2)..S<n| 00000210 3e 3d 28 6e 2f 32 29 28 32 61 2b 28 6e 2d 31 29 |>=(n/2)(2a+(n-1)| 00000220 64 29 0a 46 6f 72 20 61 6e 20 61 72 69 74 68 6d |d).For an arithm| 00000230 65 74 69 63 20 70 72 6f 67 72 65 73 73 69 6f 6e |etic progression| 00000240 2c 20 65 67 20 31 20 33 20 35 20 37 0a 39 20 65 |, eg 1 3 5 7.9 e| 00000250 74 63 2e 0a 53 3c 6e 3e 3d 73 75 6d 20 6f 66 20 |tc..S<n>=sum of | 00000260 74 68 65 20 6e 75 6d 62 65 72 73 20 75 70 20 74 |the numbers up t| 00000270 6f 20 74 65 72 6d 20 6e 2e 0a 61 3d 73 74 61 72 |o term n..a=star| 00000280 74 69 6e 67 20 6e 75 6d 62 65 72 20 28 65 67 20 |ting number (eg | 00000290 31 29 0a 64 3d 63 6f 6d 6d 6f 6e 20 64 69 66 66 |1).d=common diff| 000002a0 65 72 65 6e 63 65 20 28 65 67 20 32 29 0a 6e 74 |erence (eg 2).nt| 000002b0 68 20 74 65 72 6d 3d 61 72 5e 28 6e 2d 31 29 0a |h term=ar^(n-1).| 000002c0 46 6f 72 20 61 20 67 65 6f 6d 65 74 72 69 63 20 |For a geometric | 000002d0 70 72 6f 67 72 65 73 73 69 6f 6e 2c 20 65 67 20 |progression, eg | 000002e0 31 20 32 20 34 20 38 20 31 36 0a 61 3d 73 74 61 |1 2 4 8 16.a=sta| 000002f0 72 74 69 6e 67 20 6e 75 6d 62 65 72 20 28 65 67 |rting number (eg| 00000300 20 31 29 0a 72 3d 63 6f 6d 6d 6f 6e 20 72 61 74 | 1).r=common rat| 00000310 69 6f 20 28 65 67 20 32 29 0a 0a 0a 53 3c 6e 3e |io (eg 2)...S<n>| 00000320 3d 61 28 31 2d 72 5e 6e 29 2f 28 31 2d 72 29 0a |=a(1-r^n)/(1-r).| 00000330 46 6f 72 20 61 20 67 65 6f 6d 65 74 72 69 63 20 |For a geometric | 00000340 70 72 6f 67 72 65 73 73 69 6f 6e 2c 20 65 67 20 |progression, eg | 00000350 31 20 32 20 34 20 38 20 31 36 0a 53 3c 6e 3e 3d |1 2 4 8 16.S<n>=| 00000360 73 75 6d 20 6f 66 20 74 68 65 20 6e 75 6d 62 65 |sum of the numbe| 00000370 72 73 20 75 70 20 74 6f 20 74 65 72 6d 20 6e 2e |rs up to term n.| 00000380 0a 61 3d 73 74 61 72 74 69 6e 67 20 6e 75 6d 62 |.a=starting numb| 00000390 65 72 20 28 65 67 20 31 29 0a 72 3d 63 6f 6d 6d |er (eg 1).r=comm| 000003a0 6f 6e 20 72 61 74 69 6f 20 28 65 67 20 32 29 0a |on ratio (eg 2).| 000003b0 0a 53 3c 69 3e 3d 61 2f 28 31 2d 72 29 0a 46 6f |.S<i>=a/(1-r).Fo| 000003c0 72 20 61 20 67 65 6f 6d 65 74 72 69 63 20 70 72 |r a geometric pr| 000003d0 6f 67 72 65 73 73 69 6f 6e 2c 20 77 68 65 6e 20 |ogression, when | 000003e0 7c 72 7c 3c 31 2c 0a 65 67 20 31 20 31 2f 32 20 ||r|<1,.eg 1 1/2 | 000003f0 31 2f 34 20 31 2f 38 20 31 2f 31 36 0a 53 3c 69 |1/4 1/8 1/16.S<i| 00000400 3e 3d 73 75 6d 20 6f 66 20 74 65 72 6d 73 20 74 |>=sum of terms t| 00000410 6f 20 69 6e 66 69 6e 69 74 79 0a 61 3d 73 74 61 |o infinity.a=sta| 00000420 72 74 69 6e 67 20 6e 75 6d 62 65 72 20 28 65 67 |rting number (eg| 00000430 20 31 29 0a 72 3d 63 6f 6d 6d 6f 6e 20 72 61 74 | 1).r=common rat| 00000440 69 6f 20 28 65 67 20 bd 29 0a 65 3d 28 31 2f 30 |io (eg .).e=(1/0| 00000450 21 29 2b 28 31 2f 31 21 29 2b 28 31 2f 32 21 29 |!)+(1/1!)+(1/2!)| 00000460 2b 28 31 2f 33 21 29 2b 28 31 2f 34 21 29 2e 2e |+(1/3!)+(1/4!)..| 00000470 2e 0a 54 68 69 73 20 69 73 20 6f 6e 65 20 77 61 |..This is one wa| 00000480 79 20 6f 66 20 63 61 6c 63 75 6c 61 74 69 6e 67 |y of calculating| 00000490 20 65 20 28 32 2e 37 31 38 29 2c 0a 77 68 65 72 | e (2.718),.wher| 000004a0 65 20 61 21 20 69 73 20 74 68 65 20 66 61 63 74 |e a! is the fact| 000004b0 6f 72 69 61 6c 20 6f 66 20 61 3a 0a 61 28 61 2d |orial of a:.a(a-| 000004c0 31 29 28 61 2d 32 29 28 61 2d 33 29 2e 2e 20 75 |1)(a-2)(a-3).. u| 000004d0 6e 74 69 6c 20 61 2d 6e 3d 31 0a 4e 6f 74 65 3a |ntil a-n=1.Note:| 000004e0 20 30 21 3d 31 0a 0a 28 31 2b 78 29 5e 6e 3d 31 | 0!=1..(1+x)^n=1| 000004f0 2b 5b 6e 78 5d 2b 5b 6e 28 6e 2d 31 29 78 b2 2f |+[nx]+[n(n-1)x./| 00000500 32 21 5d 2b 2e 2e 2e 0a 4b 6e 6f 77 6e 20 61 73 |2!]+....Known as| 00000510 20 74 68 65 20 70 6f 77 65 72 20 73 65 72 69 65 | the power serie| 00000520 73 2c 20 66 6f 72 20 65 78 70 61 6e 73 69 6f 6e |s, for expansion| 00000530 0a 6f 66 20 74 68 65 20 62 72 61 63 6b 65 74 20 |.of the bracket | 00000540 28 31 2b 78 29 5e 6e 20 77 68 65 72 65 20 6e 20 |(1+x)^n where n | 00000550 69 73 20 61 6e 79 20 72 65 61 6c 0a 6e 75 6d 62 |is any real.numb| 00000560 65 72 20 28 70 6f 73 69 74 69 76 65 2c 20 6e 65 |er (positive, ne| 00000570 67 61 74 69 76 65 2c 20 69 6e 74 65 67 65 72 20 |gative, integer | 00000580 61 6e 64 0a 6e 6f 6e 2d 69 6e 74 65 67 65 72 29 |and.non-integer)| 00000590 2e 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a |................| 000005a0 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a |................| * 000005f0 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a |...............| 000005ff
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