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Question

If unun+1=nk+Ank1+Bnk2+Cnk3+.....nk+ank1+bnk2+cnk3+....., where k is positive integer, show that the series u1+u2+u3+...... is convergent if Aa1 is positive, and divergent if Aa1 is negative or zero.

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Solution

We apply Raabe's test
limnn[unun+11]=limnn[[nk+Ank1+Bnk2+...][nk+ank1+bnk2+...]1]=limnn[nk+Ank1+Bnk2+...][nk+ank1+bnk2+...][nk+ank1+bnk2+...]=limnn[[(Aa)nk1+(Bb)nk2+...][nk+ank1+bnk2+...]]=limnnk[(Aa)+(Bb)n+...]nk[1+an+bn2+....]=(Aa)
If Aa>1Aa1>0
The series is divergent
If Aa<1Aa1<0
The series is convergent

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