Certainly! Let's work through each question step by step. I'll start with the first one and proceed to the others afterward.

### **1. Find the Taylor Series Expansion**

**Problem:** Find the Taylor series expansion and interval of convergence for the power series \(\sum_{n=0}^{\infty} \frac{(-1)^n (x-1)^n}{2^n}\).

**Solution:**

The given series is:
\[
\sum_{n=0}^{\infty} \frac{(-1)^n (x-1)^n}{2^n}
\]

This is already in the form of a Taylor series centered at \( x = 1 \). Specifically, it can be written as:
\[
\sum_{n=0}^{\infty} a_n (x-1)^n
\]
where \( a_n = \frac{(-1)^n}{2^n} \).

This series resembles the geometric series:
\[
\sum_{n=0}^{\infty} r^n = \frac{1}{1-r}, \quad \text{for } |r| < 1
\]
Here, \( r = -\frac{x-1}{2} \). So, the function represented by the series is:
\[
f(x) = \frac{1}{1 - \left(-\frac{x-1}{2}\right)} = \frac{1}{1 + \frac{x-1}{2}} = \frac{2}{3 + (x-1)}
\]

Thus, the function is:
\[
f(x) = \frac{2}{x+1}
\]

**Interval of Convergence:**

To determine the interval of convergence, we consider:
\[
\left| -\frac{x-1}{2} \right| < 1
\]
This simplifies to:
\[
\frac{|x-1|}{2} < 1 \quad \Rightarrow \quad |x-1| < 2
\]
So the interval of convergence is:
\[
-2 < x-1 < 2 \quad \Rightarrow \quad -1 < x < 3
\]
Thus, the interval of convergence is \( (-1, 3) \).

### **2. Find the Power Series**

**Problem:** Find a power series centered at \( x=0 \) for the function \( f(x) = \frac{x}{2 + x^2} \).

**Solution:**

We start by rewriting the function:
\[
f(x) = \frac{x}{2 + x^2} = \frac{x}{2(1 + \frac{x^2}{2})}
\]
Now, we can express the denominator using the geometric series expansion:
\[
\frac{1}{1-u} = \sum_{n=0}^{\infty} u^n \quad \text{for } |u| < 1
\]
Here, \( u = -\frac{x^2}{2} \), so:
\[
\frac{1}{1 + \frac{x^2}{2}} = \sum_{n=0}^{\infty} \left(-\frac{x^2}{2}\right)^n = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{2^n}
\]
Thus:
\[
f(x) = \frac{x}{2} \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{2^n} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2^{n+1}}
\]

### **3. Perform the Following Integrations**

**(i) Integration:**

\[
\int (2xy + 2) \, dx
\]

**Solution:**

We integrate each term separately:
\[
\int 2xy \, dx + \int 2 \, dx
\]
The first integral (with respect to \( x \)) is:
\[
\int 2xy \, dx = 2y \int x \, dx = 2y \cdot \frac{x^2}{2} = yx^2
\]
The second integral is:
\[
\int 2 \, dx = 2x
\]
Thus, the solution is:
\[
yx^2 + 2x + C
\]
where \( C \) is the constant of integration.

**(ii) Integration:**

\[
\int (e^{3x} + x) \, dx
\]

**Solution:**

Integrating term by term:
\[
\int e^{3x} \, dx + \int x \, dx
\]
The first integral is:
\[
\int e^{3x} \, dx = \frac{e^{3x}}{3}
\]
The second integral is:
\[
\int x \, dx = \frac{x^2}{2}
\]
So, the solution is:
\[
\frac{e^{3x}}{3} + \frac{x^2}{2} + C
\]

### **4. Convert \(\frac{d^2\theta}{dt^2} = \frac{g}{l} \sin(\theta)\) from Polar to Cartesian Form**

**Solution:**

The equation given is:
\[
\frac{d^2\theta}{dt^2} = \frac{g}{l} \sin(\theta)
\]
In polar coordinates, \( \theta \) is the angle, and \( r \) is the radius. If we let \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \), then the conversion involves expressing \( \theta \) in terms of \( x \) and \( y \). The expression \( \sin(\theta) \) in Cartesian coordinates is:
\[
\sin(\theta) = \frac{y}{\sqrt{x^2 + y^2}}
\]
So, the equation in Cartesian form is:
\[
\frac{d^2\theta}{dt^2} = \frac{g}{l} \cdot \frac{y}{\sqrt{x^2 + y^2}}
\]

Would you like to explore any of these solutions further or discuss specific details? Here are some related questions to consider:

1. How can we determine the radius of convergence for a general power series?
2. What are the conditions under which the geometric series converges?
3. How can we solve higher-order differential equations using power series?
4. What is the significance of the interval of convergence in Taylor series?
5. How do polar to Cartesian coordinate transformations work for other trigonometric functions?

**Tip:** When converting between coordinate systems (like polar to Cartesian), always keep in mind the basic trigonometric identities and relationships between the coordinates.

Delve into Taylor series expansions, power series centered at x=0, integration of functions, and converting differential equations from polar to Cartesian form. Understand the methods and formulas used in solving these mathematical problems.

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