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Suppose you want to prove the following: 1/2 + 1/4 + 1/8 + . . ....

Suppose you want to prove the following: 1/2 + 1/4 + 1/8 + . . . + 1/2 n < 1 Try to prove this “directly,” using induction. I assume your attempt will fail. Describe the difficulty you run into. Now try another approach: (a) By experimenting with small values of n, guess an exact formula for the sum. (b) Prove that your guess is true. (c) As a corollary conclude: 1/2 + 1/4 + 1/8 + . . . + 1/2 n < 1

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