Two Pwr Series Questions- relatively simple

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In summary, The conversation is about two problems related to Taylor Series. The first problem involves finding the series for log(1+x) and whether the individual components of log(1+x) will be multiplied by (1/x). The second problem involves using the Taylor Series expansion of exp to compute e and whether the solution needs to be evaluated at x=1. The expert advises that for the first problem, the power series terms will be multiplied by (1/x) and for the second problem, the solution should be evaluated at x=1. The number of terms to be kept depends on the desired accuracy and a Taylor series remainder term may be needed for rigor.
  • #1
asif zaidi
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Hello:
(at least I think they are simple !)

The 1st question is not a h/w problem. It is a question from textbook
The 2nd question is a h/w problem.

Problem1:
The series for log(1+x) is proven -1<x<=1. It then says the Taylor Series for (1/x)log(1+x) follows.
Do they mean that the individual components of log(1+x) will be multiplied by (1/x). If true, would this apply for x, x^2 etc...

Problem2:
Put e=exp(1). Use the Taylor Series expansion of exp to compute e.

Solution
When I solve this, I can get the TS for exp. Do I have to evaluate it at x=1?

Thanks

Asif
 
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  • #2
For the first one, if by components you mean power series terms, yes. For the second, yes, you have to put x=1 in. How many terms you need to keep depends on the accuracy of the answer you need. You'll need a taylor series remainder term if you need to be rigorous about it.
 

1. What is a power series?

A power series is an infinite series of the form ∑(anxn) where an and x are constants. It is used to represent functions as an infinite polynomial.

2. What is the difference between a convergent and a divergent power series?

A convergent power series is one that has a finite limit as n approaches infinity, while a divergent power series does not have a finite limit. In other words, a convergent series will approach a specific value, while a divergent series will either approach infinity or oscillate between different values.

3. How can I determine the interval of convergence for a power series?

The interval of convergence for a power series can be determined by using the ratio test. This involves taking the limit as n approaches infinity of the absolute value of (an+1/an). If this limit is less than 1, the series will converge. If it is greater than 1, the series will diverge. If the limit is exactly 1, further tests are needed to determine convergence or divergence.

4. What is the importance of the radius of convergence?

The radius of convergence is the distance from the center of a power series to the point where the series begins to diverge. It is important because it determines the interval of convergence and how accurate the series will be in approximating the function it represents. A larger radius of convergence indicates a more accurate approximation.

5. How can power series be used in real-world applications?

Power series are used in many areas of science, engineering, and mathematics to approximate complicated functions. They are particularly useful in numerical analysis and in solving differential equations. For example, they can be used to calculate the trajectory of a projectile or to model the growth of a population over time.

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