Another from this afternoon...

Think of a clever way to show that 1 + 1/4 +1/9 + 1/16 + ... is between 3/2 and 7/4...

Let

A = 1 + 1/4 + 1/9 + 1/16 + ...

B = 1/6 + 1/12 + 1/20 + 1/30 + ...

C = 1 + B

D = 1 + 1/4 + B

Then B = (1/2 -1/3) + (1/3 - 1/4) + ...
= 1/2 + (-1/3 + 1/3) + (-1/4 + 1/4) + ... (can regroup terms like this as they tend to 0)
= 1/2

=> C = 3/2, D = 7/4

A lies between C and D by comparison.

Edit: This is how I'd have done it. Did you have a cleverer way in mind?

Originally posted by Acolyte
Let

A = 1 + 1/4 + 1/9 + 1/16 + ...

B = 1/6 + 1/12 + 1/20 + 1/30 + ...

C = 1 + B

D = 1 + 1/4 + B

Then B = (1/2 -1/3) + (1/3 - 1/4) + ...
= 1/2 + (-1/3 + 1/3) + (-1/4 + 1/4) + ... (can regroup terms like this as they tend to 0)
= 1/2

=> C = 3/2, D = 7/4

A lies between C and D by comparison.

Edit: This is how I'd have done it. Did you have a cleverer way in mind?

Your way is certainly cleverer than the way the examiner wanted me to do it, which also ended up being the way I did it, funnily enough. (Your way is rather shorter anyway...)