Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₁∞ (x²ᵏ)/k

Verified step by step guidance

1. Recognize the given power series: \( \sum_{k=1}^{\infty} \frac{x^{2k}}{k} \). Notice that the powers of \( x \) are even, i.e., \( x^{2k} = (x^2)^k \). This suggests rewriting the series in terms of \( y = x^2 \).
2. Rewrite the series as \( \sum_{k=1}^{\infty} \frac{y^k}{k} \), where \( y = x^2 \). This is a known power series form related to the natural logarithm function.
3. Recall the power series expansion for the natural logarithm: \( -\ln(1 - y) = \sum_{k=1}^{\infty} \frac{y^k}{k} \) for \( |y| < 1 \). This matches the rewritten series exactly.
4. Substitute back \( y = x^2 \) into the expression to identify the function represented by the original series: \( \sum_{k=1}^{\infty} \frac{x^{2k}}{k} = -\ln(1 - x^2) \) for \( |x| < 1 \).
5. Conclude that the given power series represents the function \( -\ln(1 - x^2) \) within its radius of convergence.

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

Recognizing Known Power Series Forms

Certain standard functions have well-known power series expansions, such as the geometric series or the series for logarithmic and exponential functions. Identifying these forms helps in matching a given series to a known function by comparing coefficients and powers.

Manipulating Series to Match Standard Forms

To identify a function from a given power series, it is often necessary to rewrite or manipulate the series, such as changing indices, factoring terms, or substituting variables. This process helps transform the series into a recognizable standard form linked to a known function.