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This best-selling guide--which has offered greater than 340,000 copies due to the fact that its first publication--has been completely up-to-date all through to correspons to present complicated calculus classes. an entire and accomplished assessment of the topic, this up to date variation positive factors very important new chapters on topology and Laplace transforms and crucial new theorems, with explanatory proofs.
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Additional info for Schaum's Outline of Theory and Problems of Advanced Calculus
Example text
N¼1 Method 1: lim ðÀ1Þn 6¼ 0, in fact it doesn’t exist. , it diverges. 1 Method 2: The sequence of partial sums is 1; 1 À 1; 1 À 1 þ 1; 1 À 1 þ 1 À 1; . . , 1; 0; 1; 0; 1; 0; 1; . . Since this sequence has no limit, the series diverges. 28. 1 Let un ¼ vn þ l. so that u 1 þ u2 þ Á Á Á þ un ¼ l. n v1 þ v2 þ Á Á Á þ vn ¼ 0 if lim vn ¼ 0. 1 v 1 þ v 2 þ Á Á Á þ v n @ jv1 þ v2 þ Á Á Á þ vP j þ jvPþ1 j þ jvPþ2 j þ Á Á Á þ jvn j n n n Since lim vn ¼ 0, we can choose P so that jvn j < =2 for n > P.
7. 2. The set of numbers 1; 1=3; 1=5; 1=7; . . is an infinite sequence with nth term un ¼ 1=ð2n À 1Þ, n ¼ 1; 2; 3; . . Unless otherwise specified, we shall consider infinite sequences only. LIMIT OF A SEQUENCE A number l is called the limit of an infinite sequence u1 ; u2 ; u3 ; . . if for any positive number we can find a positive number N depending on such that jun À lj < for all integers n > N. In such case we write lim un ¼ l. 1 EXAMPLE . If un ¼ 3 þ 1=n ¼ ð3n þ 1Þ=n, the sequence is 4; 7=2; 10=3; .
By definition of nested intervals, anþ1 A an ; bnþ1 @ bn ; n ¼ 1; 2; 3; . . and lim ðan À bn Þ ¼ 0. 1 Then a1 @ an @ bn @ b1 , and the sequences fan g and fbn g are bounded and respectively monotonic increasing and decreasing sequences and so converge to a and b. To show that a ¼ b and thus prove the required result, we note that b À a ¼ ðb À bn Þ þ ðbn À an Þ þ ðan À aÞ ð1Þ jb À aj @ jb À bn j þ jbn À an j þ jan À aj ð2Þ Now given any > 0, we can find N such that for all n > N jb À bn j < =3; so that from (2), jb À aj < .