Evaluating: lim n → ∞ 1 n + 1 ( ω + ν ) (...

uplakanimkk

uplakanimkk

Answered

2022-07-15

Evaluating:
lim n 1 n + 1 ( ω + ν ) ( n + 1 ) z ν 2 F 1 ( 1 , ω + ν + 1 ; n + 2 ; 1 z )

Answer & Explanation

persstemc1

persstemc1

Expert

2022-07-16Added 18 answers

So, we have
S n = 1 n + 1 ( ω + ν ) ( n + 1 ) z ν Γ ( n + 2 ) Γ ( n ω ν + 1 ) 1 n ω + ν + 1 ( 1 + O ( 1 n ) ) = ( ω + ν ) ( n + 1 ) z ν Γ ( n + 2 ) Γ ( n ω ν + 1 ) 1 n ω + ν + 2 ( 1 + O ( 1 n ) ) = z ν ( ω + ν ) ( n + 1 ) 1 n ( 1 + O ( 1 n ) ) = ( 1 ) n + 1 z ν Γ ( ω ν ) Γ ( n ω ν ) n ( 1 + O ( 1 n ) ) .
If ω + ν is not a non-negative integer, S n will tend to infinity in absolute value and will oscillate in sign.

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