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Which of the following are Taylor series about x=1

To determine which of the given expressions represent Taylor series about \( x = 1 \), we need to recall the general form of a Taylor series expansion of a function \( f(x) \) around a point \( x = a \):

\[
f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots
\]

In this case, \( a = 1 \), so the series would be expanded around \( x = 1 \):

\[
f(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f'''(1)}{3!}(x-1)^3 + \cdots
\]

Now, let's analyze the options:

1. **Option 1:**
   \[
   \sum_{n=0}^{\infty} \frac{x^n}{n!} (x-1)^n
   \]
   This is a Taylor series expansion but around \( x = 0 \), not \( x = 1 \), as the powers of \( x \) are not shifted to \( (x-1) \).

2. **Option 2:**
   \[
   1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \text{H.O.T.}
   \]
   This series is also around \( x = 0 \), as the series starts from \( x \) rather than \( (x-1) \).

3. **Option 3:**
   \[
   \frac{1}{2} \left(3(x-1) - \frac{1}{45}(x-1)^2 + \frac{1}{90}(x-1)^3\right)
   \]
   This is in the correct form for a Taylor series expansion around \( x = 1 \), as it involves powers of \( (x-1) \).

4. **Option 4:**
   \[
   \sum_{n=0}^{\infty} \frac{(2n+1)}{n!} (x-1)^{n-1}
   \]
   While this series involves \( (x-1) \), the powers are not exactly in the form \( (x-1)^n \), so this does not represent a standard Taylor series around \( x = 1 \).

5. **Option 5:**
   \[
   1 + (x-1) + \frac{(x-1)^2}{2!} + \frac{(x-1)^3}{3!} + \text{H.O.T.}
   \]
   This is the correct form of a Taylor series expansion around \( x = 1 \).

6. **Option 6:**
   \[
   25\ln(x-1) - \frac{(x-1)^2}{2!} + (x-1)^3 + \text{H.O.T.}
   \]
   This expression is not a Taylor series because it includes the logarithmic function \( \ln(x-1) \), which isn't polynomial and doesn't fit the standard Taylor series form.

### Conclusion:
- The Taylor series about \( x = 1 \) are **Options 3 and 5**.

Would you like more details or have any other questions? Here are some related questions to explore:

1. What is the general formula for the Taylor series expansion of \( e^x \) around \( x = a \)?
2. How do you determine the radius of convergence for a Taylor series?
3. Can a function have a Taylor series expansion around any point?
4. What is the relationship between Maclaurin series and Taylor series?
5. How do you calculate the coefficients in a Taylor series?
6. What is the significance of the remainder term in a Taylor series?
7. How does the Taylor series expansion change for functions with singularities near the point of expansion?
8. How does the Taylor series differ for a multivariable function?

**Tip:** Always ensure that the function is sufficiently smooth (i.e., has enough derivatives) at the expansion point when working with Taylor series.

Explore the analysis of Taylor series expansions around x=1 with detailed explanations. Determine which series represent Taylor series around the given point through clear examples and solutions. Suitable for advanced undergraduate students and enthusiasts of mathematical series expansions.

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