Deriving the Coefficients of an Infinite Power Series

In summary, to show that a function expressed as an infinite power series has the form f(x) = f(x0) + \sum^{\infty}_{n = 1}\frac{f^{n}(x0)}{n!}(x - x0)^{}, you can differentiate the power series n times and put x = a, which simplifies to f_n(a)/n! = a_n.
  • #1
DanAbnormal
23
0

Homework Statement



Show that if a function f(x) can be expressed as an infinite power series, then it has the form

f(x) = f(x0) + [tex]\sum^{\infty}_{n = 1}[/tex][tex]\frac{f^{n}(x0)}{n!}[/tex][tex](x - x0)^{}[/tex]

Homework Equations





The Attempt at a Solution



I know that for an infinite power series:

= f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a) + [tex]\frac{f''(a)}{2!}[/tex][tex](x - a)^{2}[/tex]...

which can be simplified into the above expression. But is there any groundwork that the question asks to get to this point here? I am thinking for 6 marsk i can't just right down the two lines...
 
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  • #2
A power series has the form [itex]
f(x)= \sum^{\infty}_{n = 0}
a_n (x-a)^n[/itex]. You want to show [itex]f_n(a)/n!=a_n[/itex]. Differentiate the power series n times and put x=a.
 

What is an infinite power series?

An infinite power series is a mathematical expression that represents a function as an infinite sum of terms. It is written in the form of ∑n=0∞anxn, where an is the coefficient of the term and x is the variable.

What is the convergence of an infinite power series?

The convergence of an infinite power series refers to whether the sum of its terms approaches a finite value as the number of terms increases. It can be determined by using various tests such as the ratio test, root test, or the comparison test.

What is the radius of convergence for an infinite power series?

The radius of convergence is the distance from the center of the power series to the point where the series either converges or diverges. It can be found by using the ratio test and is represented by the symbol R.

How do you find the sum of an infinite power series?

The sum of an infinite power series can be found by using the geometric series formula or by integrating the function represented by the series. The sum can also be approximated by truncating the series at a certain term.

What are some real-world applications of infinite power series?

Infinite power series have various applications in physics, engineering, and finance. They are used to approximate functions, calculate probabilities, and model real-life phenomena such as heat distribution, electrical signals, and stock prices.

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