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) .

Short answer

True

Step-by-step solution

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.

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.

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.