Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ (xᵏ)/(2ᵏ)

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Recognize that the given series is a geometric series of the form \(\sum_{k=0}^\infty ar^k\), where \(a\) is the first term and \(r\) is the common ratio.

Identify the first term \(a\) by substituting \(k=0\) into the series: \(a = \frac{x^0}{2^0} = 1\).

Identify the common ratio \(r\) by looking at the general term: \(r = \frac{x}{2}\).

Recall the formula for the sum of an infinite geometric series when \(|r| < 1\): \(S = \frac{a}{1 - r}\).

Write the function represented by the power series as \(f(x) = \frac{1}{1 - \frac{x}{2}}\) and simplify if desired.

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Power Series Representation

A power series is an infinite sum of terms in the form a_k(x - c)^k, where a_k are coefficients and c is the center. Understanding how functions can be expressed as power series allows us to analyze and approximate functions within their radius of convergence.

Geometric Series and Its Sum

A geometric series is a power series where each term is a constant ratio times the previous term, ∑ r^k. Its sum converges to 1/(1 - r) when |r| < 1. Recognizing a power series as geometric helps identify the function it represents.

Radius and Interval of Convergence

The radius of convergence defines the interval around the center where the power series converges to a finite value. For the series ∑ (x^k)/(2^k), convergence depends on |x/2| < 1, which is essential to determine the domain where the function representation is valid.

Representing functions by power series Identify the functions represented by the following power series.∑ₖ₌₀∞ (xᵏ)/(2ᵏ)