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Chapter 7: Q 67. (page 615)

Find the values of x for which the series∑k=0∞xkconverges.

Short Answer

Expert verified

The series ∑k=0∞xkconverges only for-1<x<1.

Step by step solution

01

Step 1. Given information.

Given a series∑k=0∞xk.

02

Step 2. Find all values of x for which the series converges.

A geometric series is of the form ∑k=0∞crk for some constants c and r.

Supposer is a non-zero real number, then ∑k=0∞crkconverges to c1-rif and only if r<1.

Here, the serieslocalid="1648830940128" ∑k=0∞xkhas r=x.

For the series to converge, x<1.

Note that ify<a, it follows that-a<y<a.

It follows that ∑k=0∞xkconverges for all -1<x<1.

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Most popular questions from this chapter

For each series in Exercises 44–47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder, R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that localid="1649224052075" Rn≤10-6.

∑k=0∞e-k

Determine whether the series ∑k=0∞32kconverges or diverges. Give the sum of the convergent series.

What is the contrapositive of the implication “If A, then B"?

Find the contrapositives of the following implications:

If a divides b and b dividesc, then a divides c.

Determine whether the series ∑k=0∞2k+25k-1converges or diverges. Give the sum of the convergent series.

In Exercises 48–51 find all values of p so that the series converges.

∑k=1∞lnkkp
See all solutions

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