Problem: lim n → ∞ n ⋅ sin ⁡ ( n n + 1 π...

Addison Trujillo

Addison Trujillo

Answered

2022-07-08

Problem:
lim n n sin ( n n + 1 π ) = π

Answer & Explanation

Mekjulleymg

Mekjulleymg

Expert

2022-07-09Added 14 answers

Note that
n sin ( n n + 1 π ) = n sin ( π 1 n + 1 π ) = n sin ( 1 n + 1 π ) .
use replacement u = 1 n + 1 , then u 0 when n , also n = 1 u u . therefore
lim n n sin ( n n + 1 π ) = lim u 0 1 u u sin ( π u ) = π lim u 0 ( 1 u ) sin ( π u ) π u = π lim u 0 ( 1 u ) lim u 0 sin ( π u ) π u = π .
lim n n sin ( n n + 1 π ) = lim u 0 1 u u sin ( π u ) = π lim u 0 ( 1 u ) sin ( π u ) π u = π lim u 0 ( 1 u ) lim u 0 sin ( π u ) π u = π .

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