Question

Find the disk of convergence for each of the following complex power series. 1. $e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + ...$ [equation (8.1)] 2. $z - \frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} + ...$ 3. $1 - \frac{z^2}{3!} + \frac{z^4}{5!} - ...$ 4. $\sum_{n=0}^{\infty} z^n$ 5. $\sum_{n=0}^{\infty} (\frac{z}{2})^n$ 6. $\sum_{n=1}^{\infty} n^2 (3iz)^n$ 7. $\sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!}$ 8. $\sum_{n=1}^{\infty} \frac{z^{2n}}{(2n+1)!}$ 9. $\sum_{n=1}^{\infty} \frac{z^n}{\sqrt{n}}$ 10. $\sum_{n=1}^{\infty} \frac{(iz)^n}{n^2}$ 11. $\sum_{n=0}^{\infty} \frac{(n!)^3 z^n}{(3n)!}$ 12. $\sum_{n=0}^{\infty} \frac{(n!)^2 z^n}{(2n)!}$ 13. $\sum_{n=1}^{\infty} \frac{(z-i)^n}{n}$ 14. $\sum_{n=0}^{\infty} n(n+1)(z-2i)^n$ 15. $\sum_{n=0}^{\infty} \frac{(z-2+i)^n}{2^n}$ 16. $\sum_{n=1}^{\infty} 2^n (z+i-3)^{2n}$

          Find the disk of convergence for each of the following complex power series.
1. $e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + ...$ [equation (8.1)]
2. $z - \frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} + ...$
3. $1 - \frac{z^2}{3!} + \frac{z^4}{5!} - ...$
4. $\sum_{n=0}^{\infty} z^n$
5. $\sum_{n=0}^{\infty} (\frac{z}{2})^n$
6. $\sum_{n=1}^{\infty} n^2 (3iz)^n$
7. $\sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!}$
8. $\sum_{n=1}^{\infty} \frac{z^{2n}}{(2n+1)!}$
9. $\sum_{n=1}^{\infty} \frac{z^n}{\sqrt{n}}$
10. $\sum_{n=1}^{\infty} \frac{(iz)^n}{n^2}$
11. $\sum_{n=0}^{\infty} \frac{(n!)^3 z^n}{(3n)!}$
12. $\sum_{n=0}^{\infty} \frac{(n!)^2 z^n}{(2n)!}$
13. $\sum_{n=1}^{\infty} \frac{(z-i)^n}{n}$
14. $\sum_{n=0}^{\infty} n(n+1)(z-2i)^n$
15. $\sum_{n=0}^{\infty} \frac{(z-2+i)^n}{2^n}$
16. $\sum_{n=1}^{\infty} 2^n (z+i-3)^{2n}$

Find the disk of convergence for each of the following complex power series.
1. e^z = 1 + z + (z^2)/(2!) + (z^3)/(3!) + ... [equation (8.1)]
2. z - (z^2)/(2) + (z^3)/(3) - (z^4)/(4) + ...
3. 1 - (z^2)/(3!) + (z^4)/(5!) - ...
4. ∑n=0^∞ z^n
5. ∑n=0^∞ ((z)/(2))^n
6. ∑n=1^∞ n^2 (3iz)^n
7. ∑n=0^∞((-1)^n z^2n)/((2n)!)
8. ∑n=1^∞(z^2n)/((2n+1)!)
9. ∑n=1^∞(z^n)/(√(n))
10. ∑n=1^∞((iz)^n)/(n^2)
11. ∑n=0^∞((n!)^3 z^n)/((3n)!)
12. ∑n=0^∞((n!)^2 z^n)/((2n)!)
13. ∑n=1^∞((z-i)^n)/(n)
14. ∑n=0^∞ n(n+1)(z-2i)^n
15. ∑n=0^∞((z-2+i)^n)/(2^n)
16. ∑n=1^∞ 2^n (z+i-3)^2n

Added by Emily J.

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