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Chapter 1: Q6P (page 9)

Use the preliminary test to decide whether the following series are divergent or require further testing. Careful:Do notsay that a series is convergent; the preliminary test cannot decide this.

6.鈭�n=1鈭�n!(n+1)!

Short Answer

Expert verified

The series requires further testing.

Step by step solution

01

Use the preliminary test

Consider the series:

鈭�n=1鈭�n!(n+1)!

Use the preliminary test to see if this series diverges or converges, where an=n!(n+1)!.

So, limn鈫�鈭�an=limn鈫�鈭�n!(n+1)!.

02

Check the parameter

It is known that (n+1)!=(n+1).n!. Hence,

limn鈫�鈭�an=limn鈫�鈭�n!(n+1)!=limn鈫�鈭�1(n+1)=1鈭�=0

Because limn鈫�鈭�an=0, this means that preliminary testing is not enough and the series will need to be further tested using other methods.

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Most popular questions from this chapter

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case鈭�n=1鈭�x2n2nn2

ln1+x1-x

x1+x

Use power series to evaluate the function at the given point. Compare with computer results, using the computer to find the series, and also to do the problem without series. Resolve any disagreement in results (see Example 1).11+x4-cosx2atx=0.012 at .

Use power series to evaluate the function at the given point. Compare with computer results, using the computer to find the series, and also to do the problem without series. Resolve any disagreement in results (see Example 1).earcsinx+In(1-xe)atx=0.0003.
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