Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹

Verified step by step guidance

1. First, write the given power series explicitly: \(\sum_{k=0}^{\infty} 2^{k} x^{2k+1}\).
2. Notice that the exponent on \(x\) is \(2k+1\), which can be rewritten as \(x^{2k} \cdot x^{1} = x \cdot (x^2)^k\).
3. Rewrite the series as \(\sum_{k=0}^{\infty} 2^{k} x (x^2)^k = x \sum_{k=0}^{\infty} (2 x^2)^k\).
4. Recognize that the inner sum \(\sum_{k=0}^{\infty} (2 x^2)^k\) is a geometric series with common ratio \(r = 2 x^2\).
5. Recall the formula for the sum of a geometric series \(\sum_{k=0}^{\infty} r^k = \frac{1}{1-r}\), valid when \(|r| < 1\). Use this to express the series as \(x \cdot \frac{1}{1 - 2 x^2}\).

Key Concepts

Power Series Representation of Functions

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. Functions can often be expressed as power series within a radius of convergence, allowing complex functions to be analyzed and approximated using polynomials.

Recognizing Standard Power Series Forms

Many common functions have known power series expansions, such as geometric series, exponential, sine, and cosine. Identifying the pattern of coefficients and powers in a given series helps match it to a standard form, facilitating the identification of the represented function.

Manipulating Series Indices and Terms

Reindexing or factoring terms in a power series can simplify it to a recognizable form. This includes adjusting powers of x, extracting constants, or rewriting sums to match known series, which is essential for identifying the function represented by a given power series.