Prove by math induction
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Prove that 1 + 1/2 + 1/4 + ... + 1/(2^n) = 2 - 1/(2^n) for all natural numbers n.
Having problems with n = 1 i.e. the base case.
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The expert prove by math induction. A complete, neat and step-by-step solution is provided in the attached file.
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Prove
1 + 1/2 + 1/4 + ... + 1/(2^n) = 2 - 1/(2^n) for all natural numbers n
Answer:
First, we note that 1 + [1/2 + 1/4 + ... + 1/(2^n)] = 1 + [1 - 1/(2^n)], this means we just need to prove that
1/2 + 1/4 + ... + 1/(2^n) = 1 - 1/(2^n)
Step # LHS RHS
To Prove 1/2 + 1/4 + ... + 1/(2^n) 1 - 1/(2^n)
1 Put n = 1. Then 1/2 1 - 1/(2^1) = 1/2
2 Put n = 2. Then 1/2 + ¼ = 3/4 1 - 1/(2^2) = ...
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