Find the power series representation about the center c=0 of
(a) f(x)= 1/(1+x^2)
(b) Use part (a) to find the power series of g(x)=ln(1+x^2)
-Thank You
taking various derivatives, it is clear that
1/(1+x^2) = 1 - x^2 + x^4 - x^8 + ... for |x|<1
now, since d/dx ln(1/(1+x^2)) = 2x/(1+x^2)
2x/(1+x^2) = 2x - 2x^3 + 2x^5 - 2x^7 + ...
and
ln(1/(1+x^2)) = ∫ 2x/(1+x^2) dx = x^2 - x^4/2 + x^6/3 - x^8/4 + ...
To find the power series representation of a function, we need to express it as a sum of terms involving powers of x. Let's work on each part separately:
(a) f(x) = 1/(1+x^2)
To find the power series representation of f(x) about the center c = 0, we can use the geometric series formula:
1/(1 - u) = 1 + u + u^2 + u^3 + ...
Let's rewrite f(x) as:
f(x) = 1/(1 - (-x^2))
Now, we substitute -x^2 for u in the geometric series formula:
f(x) = 1 + (-x^2) + (-x^2)^2 + (-x^2)^3 + ...
Expanding the powers of (-x^2), we get:
f(x) = 1 - x^2 + x^4 - x^6 + ...
This is now the power series representation of f(x) about the center c = 0.
(b) g(x) = ln(1+x^2)
To find the power series representation of g(x), we can integrate the power series representation of f(x) from part (a). Let's call the power series representation of f(x) as P(x):
P(x) = 1 - x^2 + x^4 - x^6 + ...
Integrating each term of P(x), we get:
∫[P(x)dx] = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + ...
Notice that we can rewrite this series as:
g(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + ...
This is the power series representation of g(x) about the center c = 0. It is obtained by integrating the power series representation of f(x) term by term.