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Use induction to show $$1+\frac14+\frac19+...+\frac1{n^2} < 2-\frac1{n}$$

Assume $P(k-1)$:

$$1+\frac14+\frac19+...+\frac1{(k-1)^2} < 2-\frac1{k-1}$$

Show $P(k)$:

I tried to show that $2-\dfrac1{(k-1)} +\dfrac1{k^2}$ was equal to the original RHS of the equation, but that doesn't seem to be the case. Am I messing up the proof somewhere?

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up vote 3 down vote accepted

What you want is

$$2 - \frac1{k-1} + \frac1{k^2} < 2 - \frac1{k}$$

which is really

$$\frac1{k-1} - \frac1{k^2} > \frac1{k}$$

That is

$$\frac1{k-1} > \frac1{k^2} + \frac{k}{k^2}$$

But then that is nothing but

$$k^2 > k^2 - 1$$

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