Why do I get different results from two calculation methods?

Question

I am wondering what the reason for the following is

We know that ,

exponential has a taylor representation : $$exp(x)=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...$$

Using the first n terms , in R , I compute

$\exp(-10)$ and get one answer.

However, if I compute $\frac{1}{\exp(10)}$, I get a slightly different answer.

For example, when I use the first 30 terms of the Taylor Series,

I get

$\exp(-10)=0.0009703416$

but $\frac{1}{\exp(10)}=4.539993\times10^{-5}$

Once I get to a large enough number of iterations, both converge to the correct value.

So why do we get differing answers when I use the same method, only replacing by a division of 1 over?

Thanks!

Answer 1

For $\exp(-10)$, more terms have to be computed since the 30-th term $\dfrac{(-10)^{30}}{30!}\approx 0.00377$ is several times bigger than the partial sum so far, 0.0009703416.

For $\exp(10)$, all terms are positive. The 30-th term $\dfrac{(10)^{30}}{30!}\approx 0.00377$ is already substantially negligible to the partial sum so far, 22026.464.

Here is the output of the program of my Python program that computes $\exp(-10)$ by that Taylor series. The first column tells the ordinals of the terms. The second columns show the values of that terms. The third terms is the partial sums so far.

0                     1.0                     1.0
1                   -10.0                    -9.0
2                    50.0                    41.0
3     -166.66666666666666     -125.66666666666666
4      416.66666666666663                   291.0
5     -833.33333333333326     -542.33333333333326
6      1388.8888888888887      846.55555555555543
7     -1984.1269841269839     -1137.5714285714284
8      2480.1587301587297      1342.5873015873012
9     -2755.7319223985887     -1413.1446208112875
10      2755.7319223985887      1342.5873015873012
11     -2505.2108385441716     -1162.6235369568703
12      2087.6756987868098      925.05216182993945
13     -1605.9043836821616      -680.8522218522221
14      1147.0745597729726      466.22233792075053
15     -764.71637318198179     -298.49403526123126
16      477.94773323873864      179.45369797750737
17     -281.14572543455211     -101.69202745704473
18      156.19206968586229      54.500042228817563
19     -82.206352466243317     -27.706310237425754
20      41.103176233121658      13.396865995695904
21     -19.572941063391266     -6.1760750676953613
22      8.8967913924505755      2.7207163247552142
23     -3.8681701706306848     -1.1474538458754706
24      1.6117375710961186     0.46428372522064798
25    -0.64469502843844739    -0.18041130321779941
26     0.24795962632247978    0.067548323104680369
27    -0.09183689863795548   -0.024288575533275111
28    0.032798892370698378     0.00851031683742327
29   -0.011309962886447721  -0.0027996460490244501
30   0.0037699876288159102  0.00097034157979145996
31  -0.0012161250415535199 -0.00024578346176205997
32  0.00038003907548547002  0.00013425561372341999
33    -0.00011516335620772       1.90922575157e-05
34  3.3871575355210003e-05  5.2963832870910003e-05
35 -9.6775929586300006e-06  4.3286239912280003e-05
36  2.6882202662899999e-06  4.5974460178560002e-05
37 -7.2654601791999997e-07  4.5247914160649999e-05
38  1.9119632050000001e-07  4.5439110481150002e-05
39 -4.9024697569999999e-08      4.539008578359e-05
40         1.225617439e-08  4.5402341957980002e-05
41 -2.9893108300000001e-09  4.5399352647149997e-05
42  7.1174066999999995e-10  4.5400064387830001e-05
43 -1.6552109000000001e-10  4.5399898866740002e-05
44            3.761843e-11      4.539993648517e-05
45 -8.3596500000000006e-12  4.5399928125519999e-05
46             1.81732e-12      4.539992994283e-05
47 -3.8665999999999999e-13  4.5399929556169997e-05
48  8.0549999999999995e-14  4.5399929636719997e-05
49 -1.6440000000000001e-14  4.5399929620280003e-05
50  3.2899999999999998e-15  4.5399929623570001e-05
51 -6.4000000000000005e-16  4.5399929622930003e-05
52                 1.2e-16  4.5399929623049997e-05
53 -2.0000000000000001e-17      4.539992962303e-05
54                     0.0      4.539992962303e-05
55                    -0.0      4.539992962303e-05
4.539992962303116e-05

Although the partial sums becomes stable after 55-th term, not all digits of 4.539992962303e-05 is correct, because of the precision of the terms such as the 8-th term. The actual value of $\exp(-10)$ should be something like 4.5399929762485e-05

In the end, it is faster and easier to compute $\exp(-x)$ to a specified precision using $\dfrac1{\exp(x)}$ instead of directly for $x>0$, especially when $x$ is large.