Geometric Series: Questions & Answers

[nokia8650]

http://img117.imageshack.us/img117/5258/w1vg1.th.jpg

http://img84.imageshack.us/img84/3151/w2px1.th.jpg


See above files (one is the question and one is the answer)

I can to the whole question, other than the last part - for part (f), why are we concerned with the sum to infinity of the series? I am confused as to why this is necessary.


Any help is appreciated.

Thanks

 

[tmc]

The sum that is done here is not the sum of the areas of every S_i; it is the sum whose result is the area of S_n itself. Look at A_3 in part (e), it is given as
A_3 = a^2 + 4a^2/9 + 4a^2/27
From this, you could guess that
A_4 = a^2 + 4a^2/9 + 4a^2/27 + 4a^2/81
And so on, such that A_n is such a sum. Every new term in the sum is the area of the extra little squares tacked onto the shape.

 

[nokia8650]

Hi, I understand that it is a geometric series, I was concerned with part (f). My question is why use the sum to infinity?

Thanks

 

[tiny-tim]

Hi, I understand that it is a geometric series, I was concerned with part (f). My question is why use the sum to infinity?

Hi nokia8650!

Because you want sup{Sn}, and the sequence is increasing, so you want S∞.

 

[nokia8650]

Thanks. The question says the sum to n, so shouldn't the equation be the sum ton, not the sum to inifnity?

Thanks

 

[tiny-tim]

Hi nokia8650!

Thanks. The question says the sum to n …

erm … no, it doesn't … it says "Find the smallest value of the constant S such that the area of Sn < S, for all values of n."

So you want sup{area of Sn}, which is the "area of S∞".

 

[nokia8650]

Thanks for the help. Its the wording of the question that is confusing me! So the question asks for the value of a constant which is greater than the area of the "final" square?

Thanks