Tìm \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaacM % gacaGGTbWaamWaaeaadaWcaaqaaiaaigdaaeaacaaIXaGaaiOlaiaa % iodaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaG4maiaac6cacaaI1a % aaaiabgUcaRiaac6cacaGGUaGaaiOlaiaac6cacqGHRaWkdaWcaaqa % aiaaigdaaeaacaWGUbWaaeWaaeaacaaIYaGaamOBaiabgUcaRiaaig % daaiaawIcacaGLPaaaaaaacaGLBbGaayzxaaaaaa!4CA5! \lim \left[ {\frac{1}{{1.3}} + \frac{1}{{3.5}} + .... + \frac{1}{{n\left( {2n + 1} \right)}}} \right]\)

Đặt \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 % da9maalaaabaGaaGymaaqaaiaaigdacaGGUaGaaG4maaaacqGHRaWk % daWcaaqaaiaaigdaaeaacaaIZaGaaiOlaiaaiwdaaaGaey4kaSIaai % Olaiaac6cacaGGUaGaaiOlaiabgUcaRmaalaaabaGaaGymaaqaaiaa % d6gadaqadaqaaiaaikdacaWGUbGaey4kaSIaaGymaaGaayjkaiaawM % caaaaaaaa!49AF! A = \frac{1}{{1.3}} + \frac{1}{{3.5}} + .... + \frac{1}{{n\left( {2n + 1} \right)}}\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGHsh % I3caaIYaGaamyqaiabg2da9maalaaabaGaaGOmaaqaaiaaigdacaGG % UaGaaG4maaaacqGHRaWkdaWcaaqaaiaaikdaaeaacaaIZaGaaiOlai % aaiwdaaaGaey4kaSIaaiOlaiaac6cacaGGUaGaaiOlaiabgUcaRmaa % laaabaGaaGOmaaqaaiaad6gadaqadaqaaiaaikdacaWGUbGaey4kaS % IaaGymaaGaayjkaiaawMcaaaaaaeaacqGHshI3caaIYaGaamyqaiab % g2da9iaaigdacqGHsisldaWcaaqaaiaaigdaaeaacaaIZaaaaiabgU % caRmaalaaabaGaaGymaaqaaiaaiodaaaGaeyOeI0YaaSaaaeaacaaI % XaaabaGaaGynaaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI1aaaai % abgkHiTmaalaaabaGaaGymaaqaaiaaiEdaaaGaey4kaSIaaiOlaiaa % c6cacaGGUaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacqGHsi % sldaWcaaqaaiaaigdaaeaacaaIYaGaamOBaiabgUcaRiaaigdaaaaa % baGaeyO0H4TaaGOmaiaadgeacqGH9aqpcaaIXaGaeyOeI0YaaSaaae % aacaaIXaaabaGaaGOmaiaad6gacqGHRaWkcaaIXaaaaiabg2da9maa % laaabaGaaGOmaiaad6gaaeaacaaIYaGaamOBaiabgUcaRiaaigdaaa % aabaGaeyO0H4Taamyqaiabg2da9maalaaabaGaamOBaaqaaiaaikda % caWGUbGaey4kaSIaaGymaaaaaaaa!8278! \begin{array}{l} \Rightarrow 2A = \frac{2}{{1.3}} + \frac{2}{{3.5}} + .... + \frac{2}{{n\left( {2n + 1} \right)}}\\ \Rightarrow 2A = 1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + ... + \frac{1}{n} - \frac{1}{{2n + 1}}\\ \Rightarrow 2A = 1 - \frac{1}{{2n + 1}} = \frac{{2n}}{{2n + 1}}\\ \Rightarrow A = \frac{n}{{2n + 1}} \end{array}\)

Nên \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaacM % gacaGGTbWaamWaaeaadaWcaaqaaiaaigdaaeaacaaIXaGaaiOlaiaa % iodaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaG4maiaac6cacaaI1a % aaaiabgUcaRiaac6cacaGGUaGaaiOlaiaac6cacqGHRaWkdaWcaaqa % aiaaigdaaeaacaWGUbWaaeWaaeaacaaIYaGaamOBaiabgUcaRiaaig % daaiaawIcacaGLPaaaaaaacaGLBbGaayzxaaGaeyypa0JaciiBaiaa % cMgacaGGTbWaaSaaaeaacaWGUbaabaGaaGOmaiaad6gacqGHRaWkca % aIXaaaaiabg2da9iGacYgacaGGPbGaaiyBamaalaaabaGaaGymaaqa % aiaaikdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaaaacqGH9a % qpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaac6caaaa!6006! \lim \left[ {\frac{1}{{1.3}} + \frac{1}{{3.5}} + .... + \frac{1}{{n\left( {2n + 1} \right)}}} \right] = \lim \frac{n}{{2n + 1}} = \lim \frac{1}{{2 + \frac{1}{n}}} = \frac{1}{2}.\)