Question

Test
the
convergence
of
series
Un=(n+1)^n/n^(n+1)​

Answer 1

Answer:

Using Partial Fraction Decomposition,

1n(n+1)(n+2)=An+Bn+1+Cn+21n(n+1)(n+2)=An+Bn+1+Cn+2

⟹1=A(n+1)(n+2)+Bn(n+2)+Cn(n+1)⟹1=A(n+1)(n+2)+Bn(n+2)+Cn(n+1)

⟹1=n2(A+B+C)+n(3A+2B+C)+2A⟹1=n2(A+B+C)+n(3A+2B+C)+2A

Comparing the coefficients of the different powers (namely, 0,1,20,1,2) of n,n, we get A=12,B=−1,C=12A=12,B=−1,C=12

⟹1n(n+1)(n+2)=12⋅

Answer 2

Answer:

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