# Power Series Expansion of f(x) = \frac{10}{1+100*x^2}

In summary, the function f(x) = \frac{10}{1+100*x^2} can be written as a power series f(x) = \sum_{n=0}^{\infty} C_nX^n, where the coefficients are given by C_n = (-100)^n for n > 0 and C_0 = 10. However, for the expansion to be valid, x must be less than 1/10 in absolute value. The coefficients of the odd powers of x are all zero, while the coefficients of the even powers of x follow a pattern of increasing powers of 10. When expanding the function, it is important to remember to account for the correct powers of 10 and to
the function $$f(x) = \frac{10}{1+100*x^2}$$
is represented as a power series
$$f(x) = \sum_{n=0}^{\infty} C_nX^n$$

Find the first few coefficients in the power series:
C_0 = ____
C_1 = ____
C_2 = ____
C_3 = ____
C_4 = ____

well $$f(x) = \frac{10}{1+100*x^2}$$ can be written as $$10\sum_{n=0}^{\infty} (-100x^2)^n$$

for C_0, i got 10 because if you plug in n = 0, you get 10 (which is correct).
for c_1, when i plug in n=1, i get -1000, which is incorrect.
i tried doing the same for c_2-c_4, but it keeps telling me i get the wrong answer. does anyone know why?

Im pretty sure you modeled the function incorrectly.

Write out a few terms of your expansion. What are the coefficients of the even powers of x in your expansion?

and more importantly, what are the coefficients of the odd powers of x?

i don't think i modeled it incorrectly, because there's a similar problem in the book, but maybe i made a mistake so who knows, but here are the first few terms...

10 - 1000x^2 + 10000x^4 -10000000x^6 + 1000000000x^8...

coefficients of the odd powers are zero...

but i can't seem to get the even coefficients correctly... what am i doing wrong?

Looks fine to me except that your coefficient for $x^4$ is one power of ten too small. Remember that this expansion is only valid for $|x|<1/10$ too.

## What is a power series expansion?

A power series expansion is a way of expressing a function as an infinite sum of powers of a variable. It is typically used to approximate a function that is difficult to solve analytically.

## How do you determine the coefficients in a power series expansion?

The coefficients in a power series expansion can be determined by using the formula: cn = f(n)(a)/n!, where cn is the coefficient of the nth term, f(n)(a) is the nth derivative of the function evaluated at the point a, and n! is the factorial of n.

## What is the general form of a power series expansion?

The general form of a power series expansion is ∑n=0∞ cn(x-a)n, where cn are the coefficients, x is the variable, and a is the point of expansion.

## How can power series expansions be used in calculus?

Power series expansions can be used in calculus to approximate the value of a function, to find derivatives and integrals of a function, and to solve differential equations.

## Can a power series expansion be used for any function?

No, power series expansions can only be used for functions that are infinitely differentiable at the point of expansion. Additionally, the expansion may only converge within a certain interval of x values.

Replies
2
Views
1K
Replies
9
Views
364
Replies
3
Views
1K
Replies
11
Views
2K
Replies
4
Views
1K
Replies
5
Views
809
Replies
2
Views
354
Replies
3
Views
2K
Replies
5
Views
456
Replies
4
Views
266