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bcjochim07
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Homework Statement
Is there a way to check if a power series representation of a function is correct?
bcjochim07 said:Ok f(0) = the value of the power series at x=0
The first derivative of the function is (-x^2 + 1)/(1-x)^4 so f'(0)=1
The first derivative of the power series is sum from n=1 to infinity of n^2*x^(n-1) and if I plug in x, I will get zero. What am I not understanding?
A power series representation is a mathematical expression that represents a function as an infinite sum of terms, where each term is a polynomial multiplied by a variable raised to a non-negative integer power.
Power series representations are useful in mathematics because they allow us to approximate functions and perform calculations with them, even if the functions are not easily integrable or differentiable. They also have applications in areas such as physics, engineering, and economics.
The convergence of a power series can be determined by using the ratio test, where the limit of the absolute value of the ratio of consecutive terms is taken as the number of terms approaches infinity. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive and another method must be used.
No, not every function can be represented as a power series. The function must have a continuous derivative on a specific interval and satisfy certain criteria, such as the Cauchy-Hadamard theorem, in order for a power series representation to exist.
A Taylor series is a specific type of power series, where the terms in the series are the derivatives of the function evaluated at a certain point. A power series, on the other hand, can be centered at any point and does not necessarily involve derivatives of the function.