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(1) Let xn and yn be sequences. Determine if the following statement is true or false. If true explain, if false give a counter example: limn→∞ |xn − yn| = 0 then limn→∞ xn = limn→∞ yn. (2) Does the series X∞ n=0 (−1)n converge? Explain your reasoning using the partial sum sequence. (3) (a) Let r be a fixed constant. Compute the following sum Xn k=0 r k . (Hint: Compute (1 − x)(1 + x + x 2 + . . . xn)) (b) Using the above result determine limn→∞ Xn k=0 r k . (The limit depends on r!) (c) Is the following statement true or false? limn→∞ r n converges ⇐⇒ X∞ k=0 r k converges. (4) In the previous problem you showed that (1 − x)(1 + x + x 2 + · · · + x n ) = 1 − x n+1 and hence for x 6= 1 we have 1 + x + x 2 + . . . xn = 1 − x n+1 1 − x . This problem will consider the consequences of this expression. (a) First an interlude: Using a basic argument find 1 + 2 + · · · + n. (b) Now differentiate both sides of 1 + x + x 2 + . . . xn = 1 − x n+1 1 − x for when x 6= 1 to find an identity for Xn k=1 kxk−1 . 1 (c) Using the previous identity how can you find a find an identity for Xn k=1 kxk ? (d) Apply the limx→1 to both sides to find an identity for 1 + 2 + · · · + n. (e) Now playing with the sum again how could you compute X∞ k=0 k  1 2 k ? (f) Finally propose a procedure for find a closed form for the sum Xn k=1 k 2 .

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