Tìm lim \(u_n\) biết \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa % aaleaacaWGUbaabeaakiabg2da9maaqahabaWaaSaaaeaacaaIXaaa % baWaaOaaaeaacaWGUbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaam % 4AaaWcbeaaaaaabaGaam4Aaiabg2da9iaaigdaaeaacaWGUbaaniab % ggHiLdaaaa!4396! {u_n} = \sum\limits_{k = 1}^n {\frac{1}{{\sqrt {{n^2} + k} }}} \)

Ta có  \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaWaaOaaaeaacaWGUbWaaWbaaSqabeaacaaIYaaaaOGaey4k % aSIaamOBaaWcbeaaaaGccqGH8aapdaWcaaqaaiaaigdaaeaadaGcaa % qaaiaad6gadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGRbaaleqa % aaaakiabgYda8maalaaabaGaaGymaaqaamaakaaabaGaamOBamaaCa % aaleqabaGaaGOmaaaakiabgUcaRiaaigdaaSqabaaaaOGaaiilaiaa % bccacaWGRbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaac6caca % GGUaGaaiOlaiaacYcacaWGUbaaaa!4F9B! \frac{1}{{\sqrt {{n^2} + n} }} < \frac{1}{{\sqrt {{n^2} + k} }} < \frac{1}{{\sqrt {{n^2} + 1} }},{\rm{ }}k = 1,2,...,n\) Suy ra \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % WGUbaabaWaaOaaaeaacaWGUbWaaWbaaSqabeaacaaIYaaaaOGaey4k % aSIaamOBaaWcbeaaaaGccqGH8aapcaWG1bWaaSbaaSqaaiaad6gaae % qaaOGaeyipaWZaaSaaaeaacaWGUbaabaWaaOaaaeaacaWGUbWaaWba % aSqabeaacaaIYaaaaOGaey4kaSIaaGymaaWcbeaaaaaaaa!43A3! \frac{n}{{\sqrt {{n^2} + n} }} < {u_n} < \frac{n}{{\sqrt {{n^2} + 1} }}\)

Mà \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaacM % gacaGGTbWaaSaaaeaacaWGUbaabaWaaOaaaeaacaWGUbWaaWbaaSqa % beaacaaIYaaaaOGaey4kaSIaamOBaaWcbeaaaaGccqGH9aqpciGGSb % GaaiyAaiaac2gadaWcaaqaaiaad6gaaeaadaGcaaqaaiaad6gadaah % aaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaaaleqaaaaakiabg2da9i % aaigdaaaa!47E9! \lim \frac{n}{{\sqrt {{n^2} + n} }} = \lim \frac{n}{{\sqrt {{n^2} + 1} }} = 1\) Nên \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaacM % gacaGGTbGaamyDamaaBaaaleaacaWGUbaabeaakiabg2da9iaaigda % aaa!3CA8! \lim {u_n} = 1\)