30 Jun 04
1 edit
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?
04 Jul 04
Originally posted by AcolyteYour 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...)
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?