For a small value of h, the Taylor series expansion for f(x + h) is A. \[{\text{f}}\left( {\text{x}} \right) + {\text{hf}}’\left( {\text{x}} \right) + \frac{{{{\text{h}}^2}}}{2}{\text{f}}”\left( {\text{x}} \right) + \frac{{{{\text{h}}^3}}}{3}{\text{f}}”\left( {\text{x}} \right) + \,…\,\infty \] B. \[{\text{f}}\left( {\text{x}} \right) – {\text{hf}}’\left( {\text{x}} \right) + \frac{{{{\text{h}}^2}}}{{2!}}{\text{f}}”\left( {\text{x}} \right) – \frac{{{{\text{h}}^3}}}{{3!}}{\text{f}}”\left( {\text{x}} \right) + \,…\,\infty \] C. \[{\text{f}}\left( {\text{x}} \right) + {\text{hf}}’\left( {\text{x}} \right) + \frac{{{{\text{h}}^2}}}{{2!}}{\text{f}}”\left( {\text{x}} \right) + \frac{{{{\text{h}}^3}}}{{3!}}{\text{f}}”\left( {\text{x}} \right) + \,…\,\infty \] D. \[{\text{f}}\left( {\text{x}} \right) – {\text{hf}}’\left( {\text{x}} \right) + \frac{{{{\text{h}}^2}}}{2}{\text{f}}”\left( {\text{x}} \right) – \frac{{{{\text{h}}^3}}}{3}{\text{f}}”\left( {\text{x}} \right) + \,…\,\infty \]

” option2=”\[{\text{f}}\left( {\text{x}} \right) – {\text{hf}}’\left( {\text{x}} \right) + \frac{{{{\text{h}}^2}}}{{2!}}{\text{f}}”\left( {\text{x}} \right) – \frac{{{{\text{h}}^3}}}{{3!}}{\text{f}}”\left( {\text{x}} \right) + \,…\,\infty \]” option3=”\[{\text{f}}\left( {\text{x}} \right) + {\text{hf}}’\left( {\text{x}} \right) + \frac{{{{\text{h}}^2}}}{{2!}}{\text{f}}”\left( {\text{x}} \right) + \frac{{{{\text{h}}^3}}}{{3!}}{\text{f}}”\left( {\text{x}} \right) + \,…\,\infty \]” option4=”\[{\text{f}}\left( {\text{x}} \right) – {\text{hf}}’\left( {\text{x}} \right) + \frac{{{{\text{h}}^2}}}{2}{\text{f}}”\left( {\text{x}} \right) – \frac{{{{\text{h}}^3}}}{3}{\text{f}}”\left( {\text{x}} \right) + \,…\,\infty \]” correct=”option1″]

The correct answer is:

C. [{\text{f}}\left( {\text{x}} \right) + {\text{hf}}’\left( {\text{x}} \right) + \frac{{{{\text{h}}^2}}}{{2!}}{\text{f}}”\left( {\text{x}} \right) + \frac{{{{\text{h}}^3}}}{{3!}}{\text{f}}”\left( {\text{x}} \right) + \,…\,\infty ]

The Taylor series expansion for a function $f$ about a point $x$ is given by:

$$f(x+h) = f(x) + hf'(x) + \frac{h^2}{2!}f”(x) + \frac{h^3}{3!}f”'(x) + \cdots$$

The first term, $f(x)$, is the value of the function at the point $x$. The second term, $hf'(x)$, is the product of the function value and the change in $x$. The third term, $\frac{h^2}{2!}f”(x)$, is the product of the change in $x$ and the second derivative of the function. The fourth term, $\frac{h^3}{3!}f”'(x)$, is the product of the change in $x$ and the third derivative of the function. And so on.

The Taylor series expansion is a way of approximating the value of a function at a point $x+h$ by using the values of the function and its derivatives at the point $x$. The more terms of the series that are included, the more accurate the approximation will be.

In the case of the Taylor series expansion for $f(x+h)$, the first term is the value of the function at the point $x$. The second term is the product of the function value and the change in $x$. The third term is the product of the change in $x$ and the second derivative of the function. The fourth term is the product of the change in $x$ and the third derivative of the function. And so on.

The Taylor series expansion is a powerful tool that can be used to approximate the value of a function at a point that is not easily evaluated. It can also be used to find the derivatives of a function.