Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)


∑ₖ₌₀∞(√x − 2)ᵏ

Verified step by step guidance

1. Recognize that the given series is a geometric series of the form \(\sum_{k=0}^\infty r^k\), where the common ratio \(r\) is \(\sqrt{x} - 2\).
2. Recall that a geometric series \(\sum_{k=0}^\infty r^k\) converges to the function \(\frac{1}{1-r}\) when \(|r| < 1\).
3. Write the function represented by the series as \(f(x) = \frac{1}{1 - (\sqrt{x} - 2)}\).
4. Simplify the denominator to get \(f(x) = \frac{1}{1 - \sqrt{x} + 2} = \frac{1}{3 - \sqrt{x}}\).
5. Determine the interval of convergence by solving the inequality \(|\sqrt{x} - 2| < 1\). This involves considering the values of \(x\) for which the absolute value condition holds, and also ensuring \(x \geq 0\) since \(\sqrt{x}\) is defined for non-negative \(x\).

Key Concepts

Geometric Series

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. It converges to a finite value if the absolute value of the ratio is less than 1, and its sum is given by S = a / (1 - r), where a is the first term and r is the ratio.

Interval of Convergence

The interval of convergence is the set of all x-values for which a given series converges. For power or related series, it is found by applying convergence tests, often involving inequalities on the variable to ensure the series terms approach zero.

Manipulating Series with Functions

To find the function represented by a series, one must recognize the pattern of terms and rewrite the series in a closed form. This often involves identifying the series type and substituting expressions (like √x − 2) as the variable to simplify and find the sum function.