We are given the series:

$\sum_{n=0}^{\infty} \frac{(n + 5i)}{(2n)!} \cdot (z - i)^n$

Step 1: Identify the General Form

This is a power series of the form:

$\sum_{n=0}^{\infty} a_n (z - i)^n$

where the coefficients $a_n$ are given by:

$a_n = \frac{n + 5i}{(2n)!}$

Step 2: Find the Radius of Convergence

To find the radius of convergence $R$, we can use the Ratio Test. The Ratio Test gives the radius of convergence using the formula:

$\frac{1}{R} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

Now, let's compute the ratio $\frac{a_{n+1}}{a_n}$:

$a_{n+1} = \frac{(n + 1 + 5i)}{(2(n+1))!} = \frac{(n + 1 + 5i)}{(2n + 2)!}$

Thus, the ratio is:

$\frac{a_{n+1}}{a_n} = \frac{(n + 1 + 5i)}{(2n + 2)!} \cdot \frac{(2n)!}{n + 5i}$

Simplify this expression:

$\frac{a_{n+1}}{a_n} = \frac{(n + 1 + 5i)}{(n + 5i)} \cdot \frac{1}{(2n+2)(2n+1)}$

As $n \to \infty$, the term $\frac{(n + 1 + 5i)}{(n + 5i)} \to 1$, and the remaining part:

$\frac{1}{(2n+2)(2n+1)} \sim \frac{1}{4n^2}$

So:

$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{1}{4n^2} = 0$

Therefore, the radius of convergence is:

$R = \infty$

Step 3: Find the Center of Convergence

The center of the series is at the point $z = i$, because the series is written in the form $(z - i)^n$, indicating the expansion is centered around $z = i$.

Final Answer:

• The center of convergence is $z = i$.
• The radius of convergence is $R = \infty$.

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Would you like further details on the steps or any additional questions? Here are 5 related questions to explore:

1. How can we confirm the convergence of the series at $z = i$?
2. What happens if the series had been centered around a different value, say $z = 0$?
3. Can the Ratio Test be applied to other power series with different forms of coefficients?
4. How can we calculate the sum of the series for specific values of $z$?
5. What other tests can be used to determine the radius of convergence?

Tip: When working with power series, always remember that the Ratio Test works well, but sometimes other tests like the Root Test can be used if the coefficients have a more complex structure.