# How do you find a power series representation for # f(z)=z^2 # and what is the radius of convergence?

In general, no manipulation is needed to find the power series of a polynomial function, as a power series is itself essentially a polynomial of infinite degree.

As for the radius of convergence, for any real value, the above power series has a single nonzero term which is equal to the square of that value, and thus does not diverge. This means the radius of convergence is infinite.

By signing up, you agree to our Terms of Service and Privacy Policy

To find the power series representation for ( f(z) = z^2 ), we can express it as:

[ f(z) = z^2 = \sum_{n=0}^{\infty} a_n z^n ]

where ( a_n ) are the coefficients of the power series.

For ( f(z) = z^2 ), we have ( a_0 = 0 ) and ( a_1 = 0 ), and for ( n \geq 2 ), ( a_n = 0 ).

So, the power series representation for ( f(z) = z^2 ) is:

[ f(z) = \sum_{n=0}^{\infty} 0 \cdot z^n = 0 ]

The radius of convergence for this power series is infinite since the function ( f(z) = z^2 ) is analytic for all values of ( z ) in the complex plane.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find n such that |Tn(1) - e| ≤ 10^-5, where Tn is the Maclaurin polynomial for #f (x) = e^x#?
- What is the interval of convergence of #sum_1^oo [(2^n)(x^n)]/sqrt(n) #?
- How do you use a Power Series to estimate the integral #int_0^0.01sin(sqrt(x))dx# ?
- What is the Maclaurin series for #e^(-x)#?
- How do you find a power series representation for #(1+x)/(1-x) # and what is the radius of convergence?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7