The infinite geometric series ∑r=0^- (a)/((y^2-4 y+5)^r-l), where y=x^2-6 x+11 has a finite sum

step 1 In this problem, I can write an equation as y square minus 4y plus 5 p is greater than 1. Solvin

The infinite geometric series ∑r=0^- (a)/((y^2-4 y+5)^r-l),...

The infinite geometric series $\sum_{\mathrm{r}=0}^{-} \frac{\mathrm{a}}{\left(\mathrm{y}^{2}-4 \mathrm{y}+5\right)^{\mathrm{r}-\mathrm{l}}}$, where $\mathrm{y}=\mathrm{x}^{2}-6 \mathrm{x}+11$ has a finite sum when $\mathrm{x}$ takes a value
other than
(a) 3
(b) $-1$
(c) 4
(d) $-3$

Key Concepts

Algebraic Substitution

Algebraic substitution is the process of replacing one variable with an equivalent expression in another variable. This technique is especially important in problems where relationships between variables are given, allowing one to express a complex condition in simpler terms. It is a powerful tool for simplifying problems, particularly when determining critical values or testing conditions such as convergence.

Infinite Geometric Series

An infinite geometric series is a sum of the form a + ar + ar² + ... that converges to a finite value if the absolute value of the common ratio, r, is less than one. The convergence condition |r| < 1 is crucial because it ensures that the successive terms diminish and the infinite sum approaches a specific limit, typically calculated as a/(1-r).

Convergence Criteria

The convergence criteria for an infinite series involve checking that the common ratio's absolute value is less than one. This condition, |r| < 1, must be satisfied to ensure the series' terms decrease sufficiently for the sum to settle to a finite number. Establishing this inequality is a fundamental step when determining if an infinite geometric series converges.

Quadratic Functions

Quadratic functions are expressions of the form ax² + bx + c and are characterized by their parabolic graphs, which have a vertex that represents the minimum or maximum value of the function. Understanding quadratic functions, including techniques such as completing the square, allows one to determine critical values and behavior of the function, which may be essential when such expressions appear in the common ratio or other parts of a series.

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