The problem asks to find the first three non-zero terms of the Taylor series expansion for $x \tan x$.

Step-by-Step Solution:

1. Recall the Taylor series expansion of $\tan x$:

The Taylor series expansion for $\tan x$ around $x = 0$ is: $\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \dots$

2. Multiply by $x$:

To get $x \tan x$, multiply each term by $x$: $x \tan x = x \cdot x + x \cdot \frac{x^3}{3} + x \cdot \frac{2x^5}{15} + \dots$ $x \tan x = x^2 + \frac{x^4}{3} + \frac{2x^6}{15} + \dots$

3. Extract the first three non-zero terms:

From the expansion above, the first three non-zero terms are: $x^2, \quad \frac{x^4}{3}, \quad \frac{2x^6}{15}.$

Final Answer:

The first three non-zero terms of the Taylor series expansion for $x \tan x$ are: $x^2 + \frac{x^4}{3} + \frac{2x^6}{15}.$

Let me know if you'd like further clarification or details!

Relative Questions:

1. How do you derive the Taylor series for a general function?
2. What are the common applications of Taylor series in physics and engineering?
3. Can the Taylor series for $\tan x$ converge for all values of $x$? Why or why not?
4. How can you approximate functions using only the first few terms of their Taylor series?
5. What are the differences between Taylor and Maclaurin series?

Tip:

When working with Taylor expansions, always check the radius of convergence to ensure your series is valid for the given range of $x$.