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## Homework Statement

i=1 Sigma n (1/i2) <= 2 - (1/n)

## The Attempt at a Solution

I've done the basic step and assumption step...little stuck on the inductive step

So far I have...

show 1 + 1/4 + 1/9 + 1/16 +...+ (1/k2) + (1/(k+1)2) <= 2 - (1/k+1)

- Thread starter sbc824
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- #1

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i=1 Sigma n (1/i2) <= 2 - (1/n)

I've done the basic step and assumption step...little stuck on the inductive step

So far I have...

show 1 + 1/4 + 1/9 + 1/16 +...+ (1/k2) + (1/(k+1)2) <= 2 - (1/k+1)

- #2

SammyS

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To clarify matters:## Homework Statement

i=1 Sigma n (1/i2) <= 2 - (1/n)

## The Attempt at a Solution

I've done the basic step and assumption step...little stuck on the inductive step

So far I have...

show 1 + 1/4 + 1/9 + 1/16 +...+ (1/k2) + (1/(k+1)2) <= 2 - (1/k+1)

I take it that you need to prove (by induction) that:

[itex]\displaystyle

\sum_{i=1}^{n}\frac{1}{i^2}\le2-\frac{1}{n}\ .[/itex]

\sum_{i=1}^{n}\frac{1}{i^2}\le2-\frac{1}{n}\ .[/itex]

Is that correct?

So, you have assumed that 1 + 1/4 + 1/9 + 1/16 +...+ (1/k

and you need to show that 1 + 1/4 + 1/9 + 1/16 +...+ (1/k

Is that correct?

What have you tried, in this effort?

BTW: Please learn to use parentheses, so that your mathematical expressions say what you mean for them to say.

- #3

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Yes, that is correct...I've done equality inductive proofs, but have not encountered less than or greater than type proofs...so I'm not sure how to begin.To clarify matters:

I take it that you need to prove (by induction) that:[itex]\displaystyle

\sum_{i=1}^{n}\frac{1}{i^2}\le2-\frac{1}{n}\ .[/itex]

Is that correct?

So, you have assumed that 1 + 1/4 + 1/9 + 1/16 +...+ (1/k^{2}) ≤ 2 - (1/k) ,

and you need to show that 1 + 1/4 + 1/9 + 1/16 +...+ (1/k^{2}) + (1/(k+1)^{2}) ≤ 2 - (1/(k+1)) .

Is that correct?

What have you tried, in this effort?

BTW: Please learn to use parentheses, so that your mathematical expressions say what you mean for them to say.

- #4

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See what happens if you combine like terms and use what you already know about [itex]\sum_{i=1}^{n}\frac{1}{i^2}[/itex] to help you out.

- #5

SammyS

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Let's see:Yes, that is correct...I've done equality inductive proofs, but have not encountered less than or greater than type proofs...so I'm not sure how to begin.

You are assuming that some k,

1 + 1/4 + 1/9 + 1/16 +...+ 1/k^{2} ≤ 2 - (1/k) .

Adding 1/(k+1)

1 + 1/4 + 1/9 + 1/16 +...+ 1/k^{2} + 1/(k+1)^{2} ≤ 2 - (1/k) +1/(k+1)^{2} .

So, if you can show that 2 - (1/k) +1/(k+1)

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