Taylor Series for f(x) = (1+x)^m

In summary, the Taylor Series for f(x) = (1+x)^m about x=0 where m is a real number is 1 + mx + 1/2(m-1)mx^2 + 1/6(m-2)(m-1)mx^3 +... . The binomial series terminates when m is a non-negative integer because all the terms after the (m+1)th term will be zero. This result from (i) can be used to find the first four non-zero terms of the series for (1+x)^(-1/2).
  • #1
Grew Gore

Homework Statement


i) What is the Taylor Series for f(x) = (1+x)^m about x=0 where m is a real number?
ii) Why does this binomial series terminate when m is a non-negative integer? A
iii) Can the result to (i) be used to find the first four non-zero terms of the series for (1+x)^(-1/2)


Homework Equations

The Attempt at a Solution


I started getting: 1 + mx + 1/2(m-1)mx^2 +1/6(m-2)(m-1)mx^3 +... but am unsure if this is right and how to go on from here.
 
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  • #2
That looks correct for (i) although they may prefer you to use general summation notation rather than '+...'
Now do (ii). Hint: Which, if any, of the terms in the expansion you wrote as answer to (i) will be zero if m is a positive integer?
 

1. What is a Taylor Series for f(x) = (1+x)^m?

A Taylor Series is a mathematical representation of a function using an infinite sum of terms, each with a different degree of the independent variable x. In this case, the function is (1+x)^m, where m is a constant. The Taylor Series of this function is given by: 1 + mx + m(m-1)x^2/2! + m(m-1)(m-2)x^3/3! + ...

2. What is the purpose of a Taylor Series?

The purpose of a Taylor Series is to approximate a complex function with a simpler one. By using more terms in the series, a more accurate approximation can be achieved. This is useful for evaluating a function at a point where it is difficult to calculate directly.

3. How is the Taylor Series for f(x) = (1+x)^m calculated?

The Taylor Series for (1+x)^m can be derived using the Taylor Series formula:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
where a is the point at which the series is centered and f'(a), f''(a), etc. are the derivatives of the function evaluated at a. By substituting in the values for (1+x)^m and its derivatives, we get the Taylor Series mentioned in the first question.

4. What is the convergence of the Taylor Series for f(x) = (1+x)^m?

The Taylor Series for (1+x)^m has a radius of convergence of 1, meaning it converges for all values of x within a distance of 1 from the center point. This means that the series will only give an accurate approximation of the function for values of x within this range. Outside of this range, the series will diverge and not accurately represent the function.

5. How is the Taylor Series for f(x) = (1+x)^m used in real-world applications?

The Taylor Series for (1+x)^m is used in many areas of physics and engineering, particularly in the fields of mechanics and electromagnetism. It is also used in finance and economics to model growth and compound interest. In computer science, the Taylor Series is used in algorithms for numerical analysis and optimization.

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