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sheet2

# sheet2 - contain sentences explaining what you are doing...

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CM221A ANALYSIS I Exercise Sheet 2 1. Is the following statement correct: if lim n →∞ a n = 1 then lim n →∞ ( n a n ) = n ? 2. Find lim n →∞ (sin n ) 2 - n . 3. Evaluate lim n →∞ n + 1 - n . Hint: use x 2 - y 2 = ( x + y )( x - y ). 4. Find lim n →∞ c 2 n - 1 c 2 n + 1 ! for all real values of c . Look separately at | c | > 1 , | c | < 1 and | c | = 1. 5. Find out which of the following series converge. Your solutions must
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Unformatted text preview: contain sentences explaining what you are doing and what tests you are using. (i) ∞ X n =1 2 n-n 3 n + n 2 (ii) ∞ X n =1 n n n ! (iii) ∞ X n =1 (-1) n n k 1 + 2 n (iv) ∞ X n =0 ( n-1) ( x + 1) n ( n + 1) 2 n (consider all possible values of x ∈ R ). 1...
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