高數(shù)級數(shù)習(xí)題課

高數(shù)級數(shù)理論部分練習(xí)10．5．28一．填空1．級數(shù)SKIPIF1<0的和為。2．把函數(shù)SKIPIF1<0展開成SKIPIF1<0的冪級數(shù)到：SKIPIF1<0。3．級數(shù)SKIPIF1<0的和為。4．設(shè)SKIPIF1<0是以SKIPIF1<0為周期的周期函數(shù)，在SKIPIF1<0上的表達式為SKIPIF1<0，則在SKIPIF1<0處SKIPIF1<0的傅里葉級數(shù)收斂于。5．冪級數(shù)SKIPIF1<0的收斂區(qū)間為。6．若冪級數(shù)SKIPIF1<0在SKIPIF1<0時收斂，則冪級數(shù)SKIPIF1<0在SKIPIF1<0時是否絕對收斂？7．若SKIPIF1<0，則級數(shù)SKIPIF1<0收斂，對么？（）8．若SKIPIF1<0在SKIPIF1<0上是以SKIPIF1<0為周期的按段光滑函數(shù)，則SKIPIF1<0=SKIPIF1<09．SKIPIF1<0，當(dāng)SKIPIF1<0=時收斂。10．級數(shù)SKIPIF1<0的部分和數(shù)列SKIPIF1<0有界，則級數(shù)SKIPIF1<0收斂。（）11．若級數(shù)SKIPIF1<0與SKIPIF1<0都發(fā)散，則SKIPIF1<0也發(fā)散。（）12．若級數(shù)SKIPIF1<0發(fā)散，則SKIPIF1<0。（）13．若級數(shù)SKIPIF1<0收斂，那么它的更序級數(shù)一定收斂。（）14．若SKIPIF1<0在SKIPIF1<0上收斂于SKIPIF1<0且每個SKIPIF1<0都在SKIPIF1<0上連續(xù)，則SKIPIF1<0也在SKIPIF1<0上連續(xù)。（）二．選擇題1．下列級數(shù)中收斂的是（）（A）SKIPIF1<0（B）SKIPIF1<0（C）SKIPIF1<0（D）SKIPIF1<0。2．若級數(shù)SKIPIF1<0收斂，則下列級數(shù)中（）收斂。（A）SKIPIF1<0（B）SKIPIF1<0（C）SKIPIF1<0（D）SKIPIF1<0。3．設(shè)SKIPIF1<0，則下列級數(shù)中和不是1的為（）（A）SKIPIF1<0（B）SKIPIF1<0（C）SKIPIF1<0（D）SKIPIF1<04．將函數(shù)SKIPIF1<0展開成SKIPIF1<0的冪級數(shù)得到（）（A）SKIPIF1<0（B）SKIPIF1<0（C）SKIPIF1<0（D）SKIPIF1<05．下列級數(shù)條件收斂的是（）（A）SKIPIF1<0（B）SKIPIF1<0（C）SKIPIF1<0（D）SKIPIF1<06．SKIPIF1<0（）A、絕對收斂B、條件收斂C、發(fā)散D、可收斂也可能發(fā)散7．SKIPIF1<0的收斂域為（）A、SKIPIF1<0B、SKIPIF1<0C、SKIPIF1<0D、SKIPIF1<08．下列級數(shù)中條件收斂的是（）A、SKIPIF1<0B、SKIPIF1<0C、SKIPIF1<0D、SKIPIF1<09．若級數(shù)SKIPIF1<0和SKIPIF1<0都發(fā)散，則（）A、SKIPIF1<0必發(fā)散；B、SKIPIF1<0發(fā)散；C、SKIPIF1<0必發(fā)散D以上說法都不對10．SKIPIF1<0是級數(shù)SKIPIF1<0收斂的。必要條件；B、充分條件；C、充要條件；D、既非充分又非必要。11．下列命題正確的是．若SKIPIF1<0與SKIPIF1<0都發(fā)散，則SKIPIF1<0也發(fā)散．若SKIPIF1<0收斂，而SKIPIF1<0發(fā)散，則SKIPIF1<0必發(fā)散．若SKIPIF1<0SKIPIF1<0…SKIPIF1<0且SKIPIF1<0絕對收斂，則SKIPIF1<0必收斂．級數(shù)SKIPIF1<0收斂的充分必要條件是它的部分和數(shù)列有界．12．下列命題正確的是．(A)絕對收斂級數(shù)的更序級數(shù)一定收斂．(B)若SKIPIF1<0為條件收斂級數(shù)，則SKIPIF1<0一定發(fā)散．(C)若SKIPIF1<0發(fā)散，則SKIPIF1<0．(D)若SKIPIF1<0收斂，則SKIPIF1<0也收斂．三．計算與證明1．求冪級數(shù)SKIPIF1<0的收斂區(qū)間及和函數(shù)。2．討論SKIPIF1<0在SKIPIF1<0時的斂散性。3．設(shè)SKIPIF1<0的傅里葉級數(shù)為SKIPIF1<0，求系數(shù)SKIPIF1<0。4．求冪級數(shù)SKIPIF1<0的收斂區(qū)間與收斂半徑。5．求冪級數(shù)SKIPIF1<0的收斂區(qū)間和收斂半徑。6判別級數(shù)SKIPIF1<0斂散性。7．求冪級數(shù)SKIPIF1<0的收斂區(qū)間與和函數(shù)。8．求冪級數(shù)SKIPIF1<0的收斂區(qū)間及和函數(shù)。9．求級數(shù)SKIPIF1<0的收斂域，并求出它的和函數(shù)，由此求出SKIPIF1<0的和。10．將SKIPIF1<0，在SKIPIF1<0上展開成余弦級數(shù)，并求出它的和函數(shù)。11．確定級數(shù)SKIPIF1<0的收斂域，并求和函數(shù)。12．求級數(shù)SKIPIF1<0在其收斂域SKIPIF1<0中的和函數(shù)。13．求級數(shù)SKIPIF1<0的收斂半徑及和函數(shù)14.求SKIPIF1<0的和函數(shù)。15．將函數(shù)SKIPIF1<0在SKIPIF1<0上展開為正弦級數(shù)、余弦級數(shù)。16．求冪級數(shù)SKIPIF1<0的和函數(shù)。參考答案填空1．SKIPIF1<02.SKIPIF1<03、04、35、SKIPIF1<06、絕對收斂7．×8．SKIPIF1<09、SKIPIF1<0；10．×；11×12×13√14×二．選擇1、C2、B3、C4、B5、A6.B；7．B；8.B；9、C；10、A；11．B12．A三．計算與證明1.SKIPIF1<0當(dāng)SKIPIF1<0時，級數(shù)成為SKIPIF1<0發(fā)散，所以收斂區(qū)間為SKIPIF1<0。SKIPIF1<0。2.當(dāng)SKIPIF1<0時，SKIPIF1<0,所以級數(shù)發(fā)散。當(dāng)SKIPIF1<0時，SKIPIF1<0，所以級數(shù)發(fā)散。當(dāng)SKIPIF1<0時，SKIPIF1<0，而SKIPIF1<0在SKIPIF1<0時收斂，所以SKIPIF1<0時SKIPIF1<0收斂。3.SKIPIF1<0SKIPIF1<0。4.SKIPIF1<0，當(dāng)SKIPIF1<0和SKIPIF1<0時級數(shù)收斂，所以收斂區(qū)間為SKIPIF1<0。收斂半徑為SKIPIF1<0。5.SKIPIF1<0，當(dāng)SKIPIF1<0時，SKIPIF1<0收斂，當(dāng)SKIPIF1<0時，SKIPIF1<0發(fā)散，所以收斂區(qū)間為SKIPIF1<0，收斂半徑為SKIPIF1<0。6.因為SKIPIF1<0，且當(dāng)SKIPIF1<0時，SKIPIF1<0，而SKIPIF1<0收斂，所以SKIPIF1<0收斂。7.SKIPIF1<0，收斂區(qū)間為SKIPIF1<0。SKIPIF1<0。8.SKIPIF1<0，而當(dāng)SKIPIF1<0時，級數(shù)SKIPIF1<0都收斂，所以收斂區(qū)間為SKIPIF1<0。令SKIPIF1<0=SKIPIF1<0，SKIPIF1<0,則SKIPIF1<0，于是SKIPIF1<0，當(dāng)x=0時，和函數(shù)為0；當(dāng)x=1時，和函數(shù)為1。9．SKIPIF1<0∴收斂域為SKIPIF1<0SKIPIF1<0令SKIPIF1<0SKIPIF1<010．SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0~SKIPIF1<011、SKIPIF1<0SKIPIF1<0SKIPIF1<0收斂域為：SKIPIF1<0SKIPIF1<012.SKIPIF1<0=SKIPIF1<0=SKIPIF1<0=SKIPIF1<0SKIPIF1<0SKIPIF1<0令SKIPIF1<0SKIPIF1<0，SKIPIF1<0SKIPIF1<0 SKIPIF1<0SKIPIF1<0，則SKIPIF1<0=SKIPIF1<0SKIPIF1<0 SKIPIF1<0=SKIPIF1<013.公比SKIPIF1<0，一般項SKIPIF1<0SKIPIF1<0；SKIPIF1<0SKIPIF1<0……10分SKIPIF1<014.SKIPIF1<0SKIPIF1<0從而收斂域為SKIPIF1<0設(shè)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0 當(dāng)SKIPIF1<0時，有SKIPIF1<0SKIPIF1<0