It looks like you've uploaded a photo containing a math exercise. Let's break it down:

Pregunta No 1: Determine the nature of the following series:

a) $\sum_{n=1}^{\infty} \frac{2^{2n}}{2n!}$

b) $\sum_{n=1}^{\infty} (-1)^n \frac{\ln(\ln n)}{\sqrt{n}}$

Pregunta No 2: Find the interval of convergence of the following power series:

$\sum_{n=0}^{\infty} \frac{n!}{4^n} (x-6)^n$

Pregunta No 3: Solve the following differential equations:

a) $(xy + 2y)dx + (x \ln y + x)dy = 0$

b) $(20x^3 - y)dx + (x + 30x^2y)dy = 0$

Pregunta No 4: Solve the following differential equation:

$y'' + 4y' + 4y + 5y = x + 2$

Would you like me to start solving any of these problems or explain the steps for any particular one? Let me know which part you'd like to focus on!

Here are some related questions you might find interesting:

1. How do you determine the convergence of series using the ratio test?
2. What methods can be used to solve linear ordinary differential equations?
3. How can we classify power series based on their radius of convergence?
4. What techniques are available for solving nonlinear differential equations?
5. How do we find the general solution of a second-order differential equation?

Tip: For series convergence, using the ratio or root test can simplify the analysis when factorials or exponentials are involved!