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Homework Statement
prove by mathematical induction that it is true:
\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+...+\frac{1}{2^{n}} < 1
carltouss619 said:Is this prove by induction: like first proving that the statement is true for n=1
then prove its true for n+1?
carltouss619 said:Is this prove by induction: like first proving that the statement is true for n=1
then prove its true for n+1?
But isn't he supposed to prove by induction? And that's the general definition of induction that I know of. But I do see where you're coming from though.Mark44 said:Not quite. First, assume that the statement is true for n = k. Then if the statement is true for n = k, show that it must also be true for n = k + 1.
In the context of the OP's problem,
a) assume the statement is true for n = k. IOW, that
\frac{1}{2} + \frac{1}{{2}^2} + ... + \frac{1}{{2}^k} < 1
b) Now prove that
\frac{1}{2} + \frac{1}{{2}^2} + ... + \frac{1}{{2}^k} + \frac{1}{{2}^k+1}< 1
It's not immediately obvious what you need to do to carry out the proof, unless you can come up with a different expression for the sum of the k terms, the expression on the left in a).
As a hint, look at the sequence of sums for
1/2
1/2 + 1/4
1/2 + 1/4 + 1/8
and so on, and see if you can discover a pattern that you can extend to the sum with k terms in it. If you can do that, you can prove the inequality marked (*).
Mark