級數(shù)求和的常用方法

1、1.7方程式法...............................................................................................31.8原級數(shù)轉(zhuǎn)化為子序列求和.....................................................................31.9數(shù)項(xiàng)級數(shù)化為函數(shù)項(xiàng)級數(shù)求和.................

2、.............................................31.10化數(shù)項(xiàng)級數(shù)為積分函數(shù)求原級數(shù)和.....................................................41.11三角型數(shù)項(xiàng)級數(shù)轉(zhuǎn)化為復(fù)數(shù)系級數(shù).....................................................41.12構(gòu)造函數(shù)計(jì)算級數(shù)和...............

3、............................................................51.13級數(shù)討論其子序列..............................................................................51.14裂項(xiàng)法求級數(shù)和.........................................................

4、.........................61.15裂項(xiàng)分拆組合法................................................................................71.16夾逼法求解級數(shù)和..............................................................................72函數(shù)項(xiàng)級數(shù)求和...

5、...........................................................................................82.1方程式法...............................................................................................82.2積分型級數(shù)求和...............

6、.....................................................................82.3逐項(xiàng)求導(dǎo)求級數(shù)和................................................................................92.4逐項(xiàng)積分求級數(shù)和...............................................

7、.................................92.5將原級數(shù)分解轉(zhuǎn)化為已知級數(shù)............................................................102.6利用傅里葉級數(shù)求級數(shù)和...................................................................102.7三角級數(shù)對應(yīng)復(fù)數(shù)求級數(shù)和..........

8、......................................................112.8利用三角公式化簡級數(shù).......................................................................122.9針對2.7的延伸...................................................................

9、...............122.10添加項(xiàng)處理系數(shù)................................................................................122.11應(yīng)用留數(shù)定理計(jì)算級數(shù)和.................................................................132.12利用Beta函數(shù)求級數(shù)和............

10、.......................................................14參考文獻(xiàn)..........................................................................................................151級數(shù)求和的常用方法級數(shù)要首先考慮斂散性，但本文以級數(shù)求和為中心，故涉及的級數(shù)均收斂且不過多討論級數(shù)斂

11、散性問題.由于無窮級數(shù)求和是個無窮問題，我們只能得到一個的極限和.加之級數(shù)能求和的本身就n??困難，故本文只做一些特殊情況的討論，而無級數(shù)求和的一般通用方法，各種方法主要以例題形式給出，以期達(dá)到較高的事實(shí)性.1數(shù)項(xiàng)級數(shù)求和1.1等差級數(shù)求和等差級數(shù)為簡單級數(shù)類型，通過比較各項(xiàng)得到其公差，并運(yùn)用公式可求和.，其中為首項(xiàng)，為公差11((1)22nnaannsnad?????）1ad證明：①，②12=...nsaaa21s=...naaa①②

12、得：??12112(...()nnnsaaaaaa??）因?yàn)榈炔罴墧?shù)11...nnaaaa???所以此證明可導(dǎo)出一個方法“首尾相加法”見1.2.1(2nnaas??）1.2首尾相加法此類型級數(shù)將級數(shù)各項(xiàng)逆置后與原級數(shù)四則運(yùn)算由首尾各項(xiàng)四則運(yùn)算的結(jié)果相同，便化為一簡易級數(shù)求和.例1:求.01235...(21)nnnnncccnc?????解：，，兩式相加得：01235...(21)nnnnnscccnc??????210(21)...5

13、3nnnnnsncccc?????，即：21012(22)(...)(1)2nnnnnnsnccccn?????????.01235...(21)(1)2nnnnnncccncn???????1.3等比級數(shù)求和等比級數(shù)為簡單級數(shù)類型，通過比較各項(xiàng)得到其公比并運(yùn)用公式可求和.當(dāng)=1，；當(dāng)≠1，，其中為首項(xiàng)，為公比.q1sna?q1(1)1naqsq???1aq證明：當(dāng)=1，易得，q1sna?當(dāng)≠1，①，②，q11111=...nsaaqa