# Taylor Series Expansions

## Elementary forms

```1/(1+x)= 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 - x^13 + x^14 - x^15 + x^16 - x^17 + x^18 - x^19 + x^20 - x^21 + x^22 - x^23 + x^24 - x^25 + x^26

sin(x)= x - 1/6 x^3 + 1/120 x^5 - 1/5040 x^7 + 1/362880 x^9 - 1/39916800 x^11 + 1/6227020800 x^13 - 1/1307674368000 x^15 + 1/355687428096000 x^17 - 1/121645100408832000 x^19 + 1/51090942171709440000 x^21

cos(x)= 1 - 1/2 x^2 + 1/24 x^4 - 1/720 x^6 + 1/40320 x^8 - 1/3628800 x^10 + 1/479001600 x^12 - 1/87178291200 x^14 + 1/20922789888000 x^16 - 1/6402373705728000 x^18 + 1/2432902008176640000 x^20

atan(x)= x - 1/3 x^3 + 1/5 x^5 - 1/7 x^7 + 1/9 x^9 - 1/11 x^11 + 1/13 x^13 - 1/15 x^15 + 1/17 x^17 - 1/19 x^19 + 1/21 x^21 - 1/23 x^23 + 1/25 x^25 - 1/27 x^27 + 1/29 x^29 - 1/31 x^31 + 1/33 x^33 - 1/35 x^35

exp(x)= 1 + x + 1/2 x^2 + 1/6 x^3 + 1/24 x^4 + 1/120 x^5 + 1/720 x^6 + 1/5040 x^7 + 1/40320 x^8 + 1/362880 x^9 + 1/3628800 x^10 + 1/39916800 x^11 + 1/479001600 x^12 + 1/6227020800 x^13 + 1/87178291200 x^14

ln(1+x)= x - 1/2 x^2 + 1/3 x^3 - 1/4 x^4 + 1/5 x^5 - 1/6 x^6 + 1/7 x^7 - 1/8 x^8 + 1/9 x^9 - 1/10 x^10 + 1/11 x^11 - 1/12 x^12 + 1/13 x^13 - 1/14 x^14 + 1/15 x^15 - 1/16 x^16 + 1/17 x^17 - 1/18 x^18
```

## Not so elementary forms

```1/sinc(2x)= 1 + 2/3 x^2 + 14/45 x^4 + 124/945 x^6 + 254/4725 x^8 + 292/13365 x^10 + 5657908/638512875 x^12 + 65528/18243225 x^14 + 33862354/23260111875 x^16 + 22998766228/38979295480125 x^18

1/sinc(x)= 1 + 1/6 x^2 + 7/360 x^4 + 31/15120 x^6 + 127/604800 x^8 + 73/3421440 x^10 + 1414477/653837184000 x^12 + 8191/37362124800 x^14 + 16931177/762187345920000 x^16 + 5749691557/2554547108585472000 x^18

1/sqrt(1+2x)= 1 - x + 3/2 x^2 - 5/2 x^3 + 35/8 x^4 - 63/8 x^5 + 231/16 x^6 - 429/16 x^7 + 6435/128 x^8 - 12155/128 x^9 + 46189/256 x^10 - 88179/256 x^11 + 676039/1024 x^12 - 1300075/1024 x^13 + 5014575/2048 x^14

1/sqrt(1+x)= 1 - 1/2 x + 3/8 x^2 - 5/16 x^3 + 35/128 x^4 - 63/256 x^5 + 231/1024 x^6 - 429/2048 x^7 + 6435/32768 x^8 - 12155/65536 x^9 + 46189/262144 x^10 - 88179/524288 x^11 + 676039/4194304 x^12

asin(asin(x))= x + 1/3 x^3 + 7/30 x^5 + 64/315 x^7 + 4477/22680 x^9 + 28447/138600 x^11 + 23029/102960 x^13 + 17905882/70945875 x^15 + 1158176431/3958416000 x^17 + 9149187845813/26398676304000 x^19

asin(x)= x + 1/6 x^3 + 3/40 x^5 + 5/112 x^7 + 35/1152 x^9 + 63/2816 x^11 + 231/13312 x^13 + 143/10240 x^15 + 6435/557056 x^17 + 12155/1245184 x^19 + 46189/5505024 x^21 + 88179/12058624 x^23

atan(atan(x))= x - 2/3 x^3 + 11/15 x^5 - 293/315 x^7 + 718/567 x^9 - 93589/51975 x^11 + 5347829/2027025 x^13 - 841018142/212837625 x^15 + 1037054989/172297125 x^17 - 1916294615267/206239658625 x^19

cos^2(x)= 1 - x^2 + 1/3 x^4 - 2/45 x^6 + 1/315 x^8 - 2/14175 x^10 + 2/467775 x^12 - 4/42567525 x^14 + 1/638512875 x^16 - 2/97692469875 x^18 + 2/9280784638125 x^20 - 4/2143861251406875 x^22

cos(sin(x))= 1 - 1/2 x^2 + 5/24 x^4 - 37/720 x^6 + 457/40320 x^8 - 389/172800 x^10 + 599/1520640 x^12 - 5497741/87178291200 x^14 + 39584029/4184557977600 x^16 - 569435807/426824913715200 x^18

cos(x)^asin(x)= 1 - 1/2 x^3 - 1/6 x^5 + 1/8 x^6 - 53/720 x^7 + 1/12 x^8 - 181/3024 x^9 + 73/1440 x^10 - 162289/3628800 x^11 + 4159/120960 x^12 - 1295893/39916800 x^13 + 133/5400 x^14 - 31521064147/1307674368000 x^15

cos(x)^atan(x)= 1 - 1/2 x^3 + 1/12 x^5 + 1/8 x^6 - 17/180 x^7 - 1/24 x^8 + 523/15120 x^9 + 73/1440 x^10 - 17063/453600 x^11 - 3989/120960 x^12 + 16279/665280 x^13 + 3161/100800 x^14 - 3618962339/163459296000 x^15

cos(x)^cos(x)= 1 - 1/2 x^2 + 7/24 x^4 - 19/180 x^6 + 1597/40320 x^8 - 373/32400 x^10 + 1673167/479001600 x^12 - 18631909/21794572800 x^14 + 690392431/2988969984000 x^16 - 9551388383/200074178304000 x^18

cos(x)^sin(x)= 1 - 1/2 x^3 + 1/8 x^6 - 1/80 x^7 - 37/1512 x^9 + 1/160 x^10 - 71/57600 x^11 + 107/24192 x^12 - 7/3520 x^13 + 1/1440 x^14 - 34298231/39626496000 x^15 + 1381/2661120 x^16 - 7891/31933440 x^17

exp(2x)= 1 + 2 x + 2 x^2 + 4/3 x^3 + 2/3 x^4 + 4/15 x^5 + 4/45 x^6 + 8/315 x^7 + 2/315 x^8 + 4/2835 x^9 + 4/14175 x^10 + 8/155925 x^11 + 4/467775 x^12 + 8/6081075 x^13 + 8/42567525 x^14 + 16/638512875 x^15

exp(asin(x))= 1 + x + 1/2 x^2 + 1/3 x^3 + 5/24 x^4 + 1/6 x^5 + 17/144 x^6 + 13/126 x^7 + 629/8064 x^8 + 325/4536 x^9 + 8177/145152 x^10 + 2665/49896 x^11 + 825877/19160064 x^12 + 12505/299376 x^13

exp(atan(x))= 1 + x + 1/2 x^2 - 1/6 x^3 - 7/24 x^4 + 1/24 x^5 + 29/144 x^6 - 1/1008 x^7 - 1219/8064 x^8 - 1163/72576 x^9 + 17321/145152 x^10 + 5465/228096 x^11 - 1867055/19160064 x^12 - 532055/19160064 x^13

exp(exp(x)-1)= 1 + x + x^2 + 5/6 x^3 + 5/8 x^4 + 13/30 x^5 + 203/720 x^6 + 877/5040 x^7 + 23/224 x^8 + 1007/17280 x^9 + 4639/145152 x^10 + 22619/1330560 x^11 + 4213597/479001600 x^12 + 27644437/6227020800 x^13

exp(sin(x))= 1 + x + 1/2 x^2 - 1/8 x^4 - 1/15 x^5 - 1/240 x^6 + 1/90 x^7 + 31/5760 x^8 + 1/5670 x^9 - 2951/3628800 x^10 - 1/3150 x^11 + 181/14515200 x^12 + 2417/48648600 x^13 + 58913/4151347200 x^14

exp(x/2)= 1 + 1/2 x + 1/8 x^2 + 1/48 x^3 + 1/384 x^4 + 1/3840 x^5 + 1/46080 x^6 + 1/645120 x^7 + 1/10321920 x^8 + 1/185794560 x^9 + 1/3715891200 x^10 + 1/81749606400 x^11 + 1/1961990553600 x^12

ln(1+ln(1+x))= x - x^2 + 7/6 x^3 - 35/24 x^4 + 19/10 x^5 - 917/360 x^6 + 8791/2520 x^7 - 97933/20160 x^8 + 310979/45360 x^9 - 211307/21600 x^10 + 2928941/207900 x^11 - 102022463/4989600 x^12

ln(cos(x))= -1/2 x^2 - 1/12 x^4 - 1/45 x^6 - 17/2520 x^8 - 31/14175 x^10 - 691/935550 x^12 - 10922/42567525 x^14 - 929569/10216206000 x^16 - 3202291/97692469875 x^18 - 221930581/18561569276250 x^20

ln(sinc(x))= -1/6 x^2 - 1/180 x^4 - 1/2835 x^6 - 1/37800 x^8 - 1/467775 x^10 - 691/3831077250 x^12 - 2/127702575 x^14 - 3617/2605132530000 x^16 - 43867/350813659321125 x^18 - 174611/15313294652906250 x^20

sec^2(x)= 1 + x^2 + 2/3 x^4 + 17/45 x^6 + 62/315 x^8 + 1382/14175 x^10 + 21844/467775 x^12 + 929569/42567525 x^14 + 6404582/638512875 x^16 + 443861162/97692469875 x^18 + 18888466084/9280784638125 x^20

sec(2x)= 1 + 2 x^2 + 10/3 x^4 + 244/45 x^6 + 554/63 x^8 + 202084/14175 x^10 + 2162212/93555 x^12 + 1594887848/42567525 x^14 + 7756604858/127702575 x^16 + 9619518701764/97692469875 x^18

sec(x)= 1 + 1/2 x^2 + 5/24 x^4 + 61/720 x^6 + 277/8064 x^8 + 50521/3628800 x^10 + 540553/95800320 x^12 + 199360981/87178291200 x^14 + 3878302429/4184557977600 x^16 + 2404879675441/6402373705728000 x^18

sinc(x)= 1 - 1/6 x^2 + 1/120 x^4 - 1/5040 x^6 + 1/362880 x^8 - 1/39916800 x^10 + 1/6227020800 x^12 - 1/1307674368000 x^14 + 1/355687428096000 x^16 - 1/121645100408832000 x^18 + 1/51090942171709440000 x^20

sin(sin(x))= x - 1/3 x^3 + 1/10 x^5 - 8/315 x^7 + 13/2520 x^9 - 47/49896 x^11 + 15481/97297200 x^13 - 15788/638512875 x^15 + 451939/126309456000 x^17 - 23252857/47517617347200 x^19

sqrt(1+2x)= 1 + x - 1/2 x^2 + 1/2 x^3 - 5/8 x^4 + 7/8 x^5 - 21/16 x^6 + 33/16 x^7 - 429/128 x^8 + 715/128 x^9 - 2431/256 x^10 + 4199/256 x^11 - 29393/1024 x^12 + 52003/1024 x^13 - 185725/2048 x^14

sqrt(1+x)= 1 + 1/2 x - 1/8 x^2 + 1/16 x^3 - 5/128 x^4 + 7/256 x^5 - 21/1024 x^6 + 33/2048 x^7 - 429/32768 x^8 + 715/65536 x^9 - 2431/262144 x^10 + 4199/524288 x^11 - 29393/4194304 x^12 + 52003/8388608 x^13

sqrt(cos(x))= 1 - 1/4 x^2 - 1/96 x^4 - 19/5760 x^6 - 559/645120 x^8 - 29161/116121600 x^10 - 2368081/30656102400 x^12 - 276580459/11158821273600 x^14 - 43947282079/5356234211328000 x^16

tan(2x)= 2 x + 8/3 x^3 + 64/15 x^5 + 2176/315 x^7 + 31744/2835 x^9 + 2830336/155925 x^11 + 178946048/6081075 x^13 + 30460116992/638512875 x^15 + 839461371904/10854718875 x^17

tan(tan(x))= x + 2/3 x^3 + 3/5 x^5 + 181/315 x^7 + 59/105 x^9 + 3455/6237 x^11 + 3332389/6081075 x^13 + 346678786/638512875 x^15 + 449509681/834978375 x^17 + 198232237033/371231385525 x^19

tan(x/2)= 1/2 x + 1/24 x^3 + 1/240 x^5 + 17/40320 x^7 + 31/725760 x^9 + 691/159667200 x^11 + 5461/12454041600 x^13 + 929569/20922789888000 x^15 + 3202291/711374856192000 x^17 + 221930581/486580401635328000 x^19

tan(x)= x + 1/3 x^3 + 2/15 x^5 + 17/315 x^7 + 62/2835 x^9 + 1382/155925 x^11 + 21844/6081075 x^13 + 929569/638512875 x^15 + 6404582/10854718875 x^17 + 443861162/1856156927625 x^19

(x+1)ln(x+1)= x + 1/2 x^2 - 1/6 x^3 + 1/12 x^4 - 1/20 x^5 + 1/30 x^6 - 1/42 x^7 + 1/56 x^8 - 1/72 x^9 + 1/90 x^10 - 1/110 x^11 + 1/132 x^12 - 1/156 x^13 + 1/182 x^14 - 1/210 x^15 + 1/240 x^16 - 1/272 x^17

(x+1)^(x+1)= 1 + x + x^2 + 1/2 x^3 + 1/3 x^4 + 1/12 x^5 + 3/40 x^6 - 1/120 x^7 + 59/2520 x^8 - 71/5040 x^9 + 131/10080 x^10 - 53/5040 x^11 + 179063/19958400 x^12 - 152587/19958400 x^13 + 1711379/259459200 x^14

(x+1)^x= 1 + x^2 - 1/2 x^3 + 5/6 x^4 - 3/4 x^5 + 33/40 x^6 - 5/6 x^7 + 2159/2520 x^8 - 209/240 x^9 + 8909/10080 x^10 - 601/672 x^11 + 18028763/19958400 x^12 - 3673/4032 x^13 + 34009847/37065600 x^14

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