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Chapter 8: Q. 4 (page 669)

What is a power series in x-x0?

Short Answer

Expert verified

A power series inx-x0is鈭�k=0鈭�akx-x0k=a0+a1x-x0+a2x-x02+...

Step by step solution

01

Step 1. Given information.

Consider the given question,

x-x0.

02

Step 2. Consider a power series.

If 鈭�k=0鈭�ak is a sequence of real numbers and x is a variable and x0be a real number. Then a power series in x-x0is a series of the form localid="1649343968159" 鈭�k=0鈭�akx-x0k.

Rewriting the series,

鈭�k=0鈭�akx-x0k=a0+a1x-x0+a2x-x02+...

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Most popular questions from this chapter

Use an appropriate Maclaurin series to find the values of the series in Exercises 17鈥�22.

鈭�k=0鈭�(-1)k蟺2k+1(2k+1)!

Prove that if x0鈭�蚁,x0+蚁is the interval of convergence for the series 鈭�k=0鈭�鈥�akx鈭�x0k, then the series converges conditionally at x0+p.

Let 鈭�k=0鈭�鈥�akxkbe a power series in x with an interval of convergence[-2,2). What is the radius of convergence of the power series 鈭�k=0鈭�鈥�ak(x鈭�3)k? Justify your answer.

Show that the power series 鈭�k=0鈭�鈥�(鈭�1)k2k+1x2k+1converges conditionally when x=1and when x=-1. What does this behavior tell you about the interval of convergence for the series?

Find the interval of convergence for power series: 鈭�k=1鈭�1k(x+2)k
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