Evaluate Limit Using Taylor Approximation of Power Series of e^h

In summary, to evaluate the limit lim (e^h-1-h-h^2/2)/h^3 as h approaches 0, we can use the four-term Taylor approximation for e^h: e^h = 1+h+h^2/2+h^3/6+h^4/24. Simplifying, we get e^h-1-h-h^2/2 = h^3/6+h^4/24... and dividing by h^3 gives us approximately 1/6. However, this is incorrect because the terms in the series should have factorials in the denominators. By correcting the series, we get the correct answer of 1/3.
  • #1
beanryu
92
0
Use a four-term Taylor approximation for e^h , for h near 0 , to evaluate the following limit.

lim (e^h-1-h-h^2/2)/h^3
h->0

i know that e^h = 1+h+h^2/2+h^3/3+h^4/4...

therefore, I say that e^h-1-h-h^2/2 = h^3/3+h^4/4...

(h^3/3+h^4/4...)/h^3 is approximately = 1/3

but its wrong

please give me some hints THANX!

sorry for posting it at precalculus... please help urgent!
 
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  • #2
It's not wrong.
 
  • #3
But It Is...
 
  • #4
Ack, you're right it is wrong. Your mistake is in the terms of the series.

i know that e^h = 1+h+h^2/2+h^3/3+h^4/4...

Nope, those denominators should have factorials in them, like so:

e^h = 1+h+h^2/2!+h^3/3!+h^4/4!
e^h = 1+h+h^2/2+h^3/6+h^4/24

That should fix it.
 

1. What is the Taylor approximation of a power series?

The Taylor approximation of a power series is a mathematical technique used to approximate a function by expressing it as an infinite sum of polynomials. It is based on the Taylor series, which is a way of representing a function as an infinite sum of terms that are calculated from the function's derivatives at a single point.

2. How is the Taylor approximation of a power series used to evaluate limits?

The Taylor approximation of a power series can be used to evaluate limits by finding a polynomial approximation of the function near the point at which the limit is being taken. This approximation can then be used to estimate the value of the function at that point, giving us an estimate of the limit.

3. What is the power series of e^h?

The power series of e^h is a special case of the Taylor series for the exponential function. It can be written as e^h = 1 + h + (h^2)/2! + (h^3)/3! + ... + (h^n)/n!, where n is a positive integer.

4. How accurate is the Taylor approximation for evaluating limits?

The accuracy of the Taylor approximation for evaluating limits depends on the function and the point at which the limit is being taken. In general, the approximation becomes more accurate as we use more terms in the series. However, for some functions, the Taylor series may not converge at all, making the approximation unreliable.

5. Are there any limitations to using the Taylor approximation for evaluating limits?

Yes, there are certain limitations to using the Taylor approximation for evaluating limits. As mentioned before, the series may not converge for some functions, making the approximation unreliable. Additionally, the Taylor approximation is only valid for points near the point at which the limit is being taken, so it may not give an accurate estimate for points that are far away from this point.

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