Functions as Power Series

In summary, the conversation discusses the use of the alternating series test to determine the convergence or divergence of a given series. The series in question is (-1)^n/10 and there is confusion about its convergence due to the alternating sign and the value of bn. The person concludes that the series does not converge as it does not approach 0.
  • #1
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Homework Statement


I'm having a little trouble concerning the part where we have to calculate the endpoints of a known interval of convergence in order to see whether they are convergent or divergent. In this case, is the summation of n=0 to infinity of (-1)^n/10 diverging or converging. Are there any tests to prove that? I'm kinda confused. Thanks


Homework Equations





The Attempt at a Solution

 
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  • #2
Have you looked at the alternating series test?
 
  • #3
Mark44 said:
Have you looked at the alternating series test?

uhh yea, but it just gives bn= 1/10 which doesn;t make sense...i think
 
  • #4
(-1)^n/10 doesn't converge. Don't be silly. It doesn't even approach 0. Did you mean to post something else?
 

1. What is a power series?

A power series is a mathematical representation of a function as an infinite sum of powers of a variable. It has the form ∑n=0∞c_nx^n, where c_n is a coefficient and x is the variable.

2. How are functions represented as power series?

Functions are represented as power series by expanding them around a particular point, usually 0, and expressing them in the form of a power series with coefficients determined by the derivatives of the function at that point.

3. What is the interval of convergence for a power series?

The interval of convergence for a power series is the range of values for the variable x for which the series converges. It is determined by the ratio test or the root test, and can be open, closed, or half-open depending on the convergence behavior of the series.

4. How can power series be used to approximate functions?

Power series can be used to approximate functions by truncating the series to a finite number of terms. This results in a polynomial function that closely approximates the original function within the interval of convergence.

5. What are some real-world applications of power series?

Power series have many practical applications in fields such as physics, engineering, and finance. They can be used to model physical phenomena, approximate complex functions in engineering problems, and calculate financial derivatives such as option prices.

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