Compute Taylor Series & Approximate Integral of Exponential Function

In summary, the problem involves computing the Taylor Series expansion of f(x) = exp(-x^2) around 0 and using it to find an approximate value of the integral (from 0 to 0.1) of exp(-t^2) dt. The solution suggests using the Taylor Series formula and evaluating it at x=0 and x=0.1, subtracting the two values to calculate the integral. However, it is recommended to calculate the integral directly using the Taylor Series formula of order 6, which results in a more accurate approximation with an error of less than 10^-10. The remaining terms can be used to estimate the error of the approximation.
  • #1
asif zaidi
56
0
Problem Statement
Compute the Taylor Series expansion of f(x) = exp(-x^2) around 0 and use it to find an approximate value of the integral (from 0 to 0.1) of exp(-t^2) dt

Solution

Part1:
First to compute the Taylor Series - I am pretty sure about this step so I will not give details. But if I am wrong, please correct.

Taylor Series = 1 - (x^2) + (x^4)/2! - (x^6)/3! + ... --- EQUATION 1

And a closed form solution is from Sum (i from 0 to inf) of (-1^i)* (x^2*i)/i!

Part2:
This is the part I am not doing right - maybe I am not approaching the problem in the right way.
To solve the integral evaluate the Taylor Series in equation 1 above at 0.1 and 0 and subtract. Also, I took just the first 4 terms of equation 1. Is there a way I can determine how many terms I should take?

At x = 0.1: Equation 1 is evaluated to 0.99004983375
At x = 0.0: Equation 1 is evaluated to 1.000000000000
Subtracting above gives me an integral value of -0.00995016625.

So my two questions are:

1- Now this I know is clearly wrong as the value should be positive but I cannot figure out what I am doing wrong. The absolute value above is right but why am I getting a negative. I tried above method for positive exponentials and it worked but any negative exponential, I am always getting the negative answer.

2- How do I determine how many terms I should use in my Taylor Series expression.


Thanks


Asif
 
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  • #2
Actually, don't you think you should rather calculate
[tex]\int_0^{0.1} T_n(x) \, \mathrm{d}x[/tex]
instead of
[tex]T_n(0.1) - T_n(0)[/tex]
where [itex]T_n(x)[/itex] is the n-th order Taylor series for the function
(so for example, [itex]T_4(x) = T_5(x) = 1 - x^2 + x^4/2[/itex] and [itex]T_6(x) = 1 - x^2 + x^4/2 - x^6/6[/itex]).

Then you can just take it up to order 6 and already get an error of less than [itex]10^{-10}[/itex]. You could do look at the terms you have neglected to estimate the error (e.g. neglecting x^8 and higher, the error you make will be at most of order 1/9 (0,1)^9).
 
  • #3
i don't know abput you. but i only got notes on these questions. cheak then for refernbce. my blog can be found at the top blogs section. kaixuan
 

What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms, where each term is a derivative of the function evaluated at a specific point.

Why is a Taylor series useful?

A Taylor series can be used to approximate the value of a function at a specific point, even if the function is not known exactly. It can also be used to evaluate the behavior of a function at different points and make predictions about its behavior.

How is a Taylor series computed?

A Taylor series is computed by taking derivatives of a function at a specific point and using those values to create the terms in the series. The more terms that are included in the series, the more accurate the approximation will be.

What is an exponential function?

An exponential function is a mathematical function in the form of f(x) = ab^x, where a and b are constants and x is a variable. The value of b determines the rate at which the function increases or decreases.

How can a Taylor series be used to approximate the integral of an exponential function?

By computing the Taylor series of an exponential function, we can approximate the value of the integral of that function at a specific point. This can be useful in solving problems in physics, engineering, and other fields where the integral of an exponential function may need to be evaluated.

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