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- Politecnico di Milano
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- Computer Engineering
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- Analisi Matematica 1
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ANALISI MATEMA TICA 1 Prof. E.Maluta SERIE NUMERICHE Determinare il carattere delle seguenti serie 1. +1X n=1 sin 1/n 2. +1X n=1 (°1)nsin 1/n 3. +1X n=1 ° cos 1 n °1 ¢ 4. +1X n=1 ° cos 1p n °1 ¢ 5. +1X n=1 (1°e p 1/n)/n 6. +1X n=1 (1 +e p 1/n)/n 7. +1X n=1 ° log(n2+ 1)°log(n2) ¢ 8. +1X n=1 ° log(n2+n)°log(n2) ¢ 9. +1X n=1 ° cos 1 n °1+ 1 2n ¢ 10. +1X n=1 ° cos 1p n °1+ 1 2n ¢ Determinare, al variare del parametro Æ2R, il carattere della serie 11. +1X n=1 (°1)n·nÆ 12. +1X n=1 nÆsin(1/n) 13. +1X n=1 nsin(1/nÆ) 14. +1X n=1 ° cos 1 nÆ°1+ 1 2n ¢ Æ∏0 15. +1X n=1 ° cos 1 nÆ°1+ 1 2 p n ¢ Æ∏0 SOLUZIONI: 1.,4.,6.,8.,9. divergono; 2.,3.,5.,7.,10. convergono. Serie telescopiche 1)∑n!1 " 1 n2"n 2)∑n!1 " 6 4n2−1 3)∑n!1 "ln!1"1n" 4)∑n!1 " n !n"1"! 5)∑n!2 "! 1 ln!n"1"− 1 ln!n"" 6)∑n!1 "!1 n2− 1 !n2"1"" 7)∑n!1 " 2 n3"3n2"2n Soluzioni 1) 1 n2"n!An" B n"1#A!1,B!−1 SN!!1−1 2""!1 2−1 3""..."!1 N− 1 N"1"→1 per N→"#la serie converge a 1 2) 6 4n2−1! A 2n−1" B 2n"1#A!3,B!−3 SN!!3−3 3""!3 3−3 5""..."! 3 2N−1− 3 2N"1"→3 per N→"#la serie converge a 3 3)ln(1"1n"!ln!n"1"−lnn SN!!ln 2−ln 1""!ln 3−ln 2""....!ln!N"1"−lnN"!ln!N"1"→"per N→"quindi la serie diverge 4)an!n"1−1 !n"1"!! n"1 !n"1"!− 1 !n"1"!! 1 !n"!− 1 !n"1"! SN!!1−1 2!""!1 2!−1 3!""...!1 N!− 1 !N"1"!"!1− 1 !N"1"!→1 per N→"quindi la serie converge a 1 5)SN!!1 ln 3−1 ln 2""!1 ln 4−1 ln 3""..."! 1 ln!N"1"−1 lnN"!−1 ln 2" 1 ln!N"1"→− per N→"quindi la serie converge a−1 ln 2 6)SN!!1−1 4""!1 4−1 9""..!1 N2− 1 !N"1"2"!1− 1 !N"1"2→1 per N→"quindi la serie converge a 1 7) 2 n3"3n2"2n!An" B n"1" C n"2#A!1,B!1,C!−2 SN!!1"1 3−2 2""!1 2"1 4−2 3""...!1 N" 1 N"1− 2 N"2"→1 2per N→"quindi la serie converge a 1/2 Serie geometriche 1)∑n!0 "!1 2"n 2)∑n!0 "3!3 2"n 3)∑n!0 "!1 3 "n 4)∑n!2 "!−2 3"n 5)∑n!0 " !−1" 22n n 6)∑n!1 " 2n 3n"1 7)∑n!0 " 2n e2n 8)∑n!1 " 2…
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