Problem 65 - Essential Calculus

Question

True or False? In Exercises \(63-66,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the interval of convergence for \(\sum_{n=0}^{\infty} a_{n} x^{n}\) is \((-1,1),\) then the interval of convergence for \(\sum_{n=0}^{\infty} a_{n}(x-1)^{n}\) is (0,2) .

Step-by-Step Solution

Answer

True

Step 1: Understand the Statement

In a power series \(\sum_{n=0}^{\infty} a_{n} x^{n}\), the interval of convergence is the set of x-values for which the series converges. If we replace \(x\) with \((x-1)\), it essentially shifts the interval of convergence to the right by 1 unit. This change doesn't widen or shorten the interval, simply shifts it.

Step 2: Apply Shift to the Original Interval

The original interval of convergence is \((-1,1)\). If we shift this interval to the right by 1 unit, we get the interval \((0,2)\). This means we're replacing each x-value in the original interval with \((x-1)\), which results in the shifted interval.

Step 3: Final Verification

Having shifted the interval appropriately, we see it matches the interval given in the statement, which is \((0,2)\). Therefore, the given statement is indeed true.

Key Concepts

Interval of Convergence

The interval of convergence of a power series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) is the set of all \(x\)-values for which the series converges. To understand this, think of it as finding the "safe zone" where the series behaves nicely, summing up to a finite number. This interval might have endpoints where the series converges or diverges. Thus, it's essential to test these endpoints to determine their status.

• This interval can be bounded, like \((-1, 1)\), meaning it covers all numbers between -1 and 1 but not the endpoints themselves.
• It is crucial to identify whether each endpoint is included (closed interval) or excluded (open interval), noted by brackets \([a, b]\) or parentheses \((a, b)\).

Understanding the interval is key because it tells you the range of \(x\)-values where your series will work effectively.

Convergence of Series

A series converges when the sum of its infinite terms results in a finite value. In the context of power series, such as \(\sum_{n=0}^{\infty} a_{n} x^{n}\), convergence means finding values of \(x\) that make the infinite sum reach a specific number, rather than running off to infinity.

• Convergence is closely related to the radius of convergence, which is the distance from the center of the series to the endpoints of the interval of convergence.
• The series converges absolutely if it converges for all points within a certain radius, and conditionally at the endpoints.

By understanding where a series converges, you not only know which values make it behave but also avoid those that lead to undefined or infinite sums.

Shifting Intervals

Shifting intervals in the context of power series involves changing the variable to translate the interval of convergence. For example, in the series \(\sum_{n=0}^{\infty} a_{n}(x-1)^{n}\), replacing \(x\) with \((x-1)\) shifts all \(x\)-values in the interval to the right by 1 unit.

• In the given example, the original interval \((-1,1)\) becomes \((0,2)\) after shifting. This doesn't change the width of the interval (both are of length 2).
• Shifting is a straightforward process but requires careful adjustment of endpoints to ensure the new interval correctly reflects convergence limits.

Through shifting, you maintain the properties of convergence without altering the relative behavior of the series. It's like moving a sliding window across the number line, keeping everything inside view but in a different position.