To determine the interval of convergence for the function [tex]\( f(x) = \frac{1}{(x+10)(x+11)} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] for which the function is defined. The function [tex]\( f(x) \)[/tex] is a rational function, and rational functions are undefined wherever their denominators are zero.
The denominator of the function is [tex]\((x+10)(x+11)\)[/tex]. We set the denominator equal to zero to find the points at which the function is undefined:
[tex]\[
(x + 10)(x + 11) = 0
\][/tex]
Solving this equation, we find the roots:
[tex]\[
x + 10 = 0 \quad \text{or} \quad x + 11 = 0
\][/tex]
[tex]\[
x = -10 \quad \text{or} \quad x = -11
\][/tex]
These roots, [tex]\( x = -10 \)[/tex] and [tex]\( x = -11 \)[/tex], are the points where the function is undefined. Therefore, the function is defined for all real numbers except at [tex]\( x = -10 \)[/tex] and [tex]\( x = -11 \)[/tex].
To express the interval of convergence, we exclude the points where the function is undefined from the real number line. In interval notation, this can be written as:
[tex]\[
(-\infty, -11) \cup (-11, -10) \cup (-10, \infty)
\][/tex]
Thus, the interval of convergence for the function [tex]\( f(x) = \frac{1}{(x+10)(x+11)} \)[/tex] is:
[tex]\[
(-\infty, -11) \cup (-11, -10) \cup (-10, \infty)
\][/tex]
