Problem 52 - Essential Calculus

Question

The radius of convergence of the power series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) is \(3 .\) What is the radius of convergence of the series \(\sum_{n=1}^{\infty} n a_{n} x^{n-1} ?\) Explain.

Step-by-Step Solution

Answer

The radius of convergence of the series \(\sum_{n=1}^{\infty} n a_{n} x^{n-1}\) is 3.

Step 1: Understanding the given information

Initially the problem provides a power series \(\sum_{n=0}^{\infty} a_{n}x^{n}\) and states that the radius of convergence is 3. It then asks about the radius of convergence of the series \(\sum_{n=1}^{\infty} n a_{n} x^{n-1}\). This new series is derived from the original series by multiplying each term by \(n\) and decreasing the power of \(x\) by 1.

Step 2: Observing the effects of the manipulations

Neither multiplying each term by \(n\) nor decreasing the power of \(x\) by 1 affects the radius of convergence. The radius of convergence is generally determined by the coefficients \(a_n\) of the series and is not affected by the indexing or the power of \(x\). Therefore, even though the second series is different due to the multiplication by \(n\) and the change of power of \(x\), the radius of convergence remains the same as in the first series.

Step 3: Result

The radius of convergence of the series \(\sum_{n=1}^{\infty} n a_{n} x^{n-1}\) is the same as the radius of convergence of the original series, which is 3.

Key Concepts

Power Series

A power series is a type of infinite series where each term is composed of a coefficient and a variable raised to a power. This can be expressed in the form: \[\sum_{n=0}^{\infty} a_{n}x^{n} \]Here, \(a_n\) represents the coefficients, and \(x\) is the variable raised to increasing powers starting from zero. Power series can take on the shape of familiar functions like exponential, trigonometric, or polynomial functions.
Understanding the behavior of power series is crucial because they help us approximate complex functions. The series converge or diverge based on certain values of \(x\), determining where the approximation holds true.

• The convergence of a power series is about finding the range of \(x\) values that make the series sum to a finite number.
• The range where it converges defines the radius of convergence.
• Outside this range, the series does not sum to a finite value and thus diverges.

Convergence Criteria

Convergence criteria for power series revolve around determining which values of \(x\) allow the series to converge. One common method for finding convergence is the Ratio Test, which examines the limit of the absolute ratio of successive terms in the series:

• If the limit is less than 1, the series converges.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

The radius of convergence is found using this test by setting the limit less than 1 and solving for \(x\). This radius defines an interval centered on zero where the series converges.
In the original problem, both the original and modified series share the same radius of convergence because transformations that change only the indices or coefficients by algebraic manipulation, like multiplying by \(n\) or adjusting the power of \(x\), do not affect the convergence characteristics determined by the coefficients \(a_n\).

Index Transformation

An index transformation involves changing the index variable in a series, effectively altering how we sum the terms. In the exercise provided, we see a change from \[\sum_{n=0}^{\infty} a_{n}x^{n} \]to \[\sum_{n=1}^{\infty} n a_{n} x^{n-1}\]This transformation involves two main changes:

• The power of \(x\) is reduced by one, shifting how terms are aligned with powers of \(x\).
• The coefficients are multiplied by \(n\), making each term more prominent as \(n\) increases.

Such transformations can seem to change the series significantly. However, they often preserve intrinsic properties like the radius of convergence, as noted in the solution.
These manipulations maintain the series' convergence because convergence is based on the behavior of the coefficients \(a_n\), which remain constant during such transformations. This insight helps mathematicians transform and manipulate series without accidentally changing fundamental properties such as the radius of convergence.