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Prove that the ratio test will be inconclusive on every series of the form k=1akwhereakis a rational function of k.

Short Answer

Expert verified

Hence, proved.

Step by step solution

01

Step 1. Given Information.

ak is a rational function of k.

02

Step 2. Ratio test.

Theratioteststatesthatifk=1akbeaseries,ifL=limkak+1ak,theni.IfL<1seriesconverges.ii.IfL=1thetestisincconlusive.iii.IfL>1seriesdiverges.

03

Step 3. General rational term.

Letr(x)=i=0aibixi,wherebi0.Theratior(x+1)r(x)=aibix+1iaibixi=x+1ixi=1+1xi.limxr(x+1)r(x)=limx1+1xi=limx1+0i=1.Thusthevalueoftheratiois1.Thereforetheratiotestbecomesinconclusive.

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