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A power series is an infinite sum of the form \(\sum c_k x^k\), where each \(c_k\) is a coefficient and \(x\) is the variable. In a power series, we see various powers of \(x\) combined with coefficients. These series resemble polynomials but extend infinitely.

Power series are used widely in mathematical analysis due to their ability to approximate functions accurately. They are particularly helpful when exact solutions cannot be obtained easily.

• Every power series has a center, usually denoted as the point where the series is centered.
• The **radius of convergence** \(R\) defines where the series converges around this center.
• The **interval of convergence** describes the actual domain over which the series converges.

In our exercise, we are given a power series \(f(x) = \sum c_k x^k\), with a known interval of convergence \(|x| < R\). This interval indicates that within these bounds, the series converges to a specific function; outside it, the behavior can vary significantly.

The substitution technique in the context of power series involves replacing a variable within the series with another expression. This allows exploring how transformations of the argument affect convergence.

In this exercise, we focus on substituting \(x - a\) in place of \(x\) in the original power series \(f(x)\). This process is key when dealing with series centered around different points.

• To substitute, replace \(x\) with \(x-a\) which results in \(\sum c_k (x-a)^k\).
• This transformation moves the series' center from \(x = 0\) to \(x = a\).
• We look at the new expression and observe how it influences the original conditions of convergence.

Through substitution, not only do we alter the appearance of the series, but we also shift its convergence profile, necessitating a re-analysis for the new interval.

Convergence analysis helps determine where a power series converges. It is crucial for understanding series behavior and for applications in approximating functions.

For the power series \(\sum c_k (x-a)^k\), convergence is determined by ensuring \(|x-a| < R\), the new interval of convergence derived from our substitution process.

This involves several important considerations:

• First, redefine the convergence interval based on \(x-a\) by converting \(|x|
• The new interval represents where the modified series converges accurately.
• Both endpoints of the interval need checking to determine whether convergence is inclusive or exclusive at the boundaries.

In this exercise, by shifting the series via substitution, we see how the endpoint seemingly moves but the range \(|x-a|