Last edited by Stanley_Marsh (2007-04-05 09:46:51)

Last edited by Stanley_Marsh (2007-04-05 09:58:11)

Testing this for the first few values of n, I find a₁ = 2, a₂ = 4, a₃ = 6, a₄ = 8, a₅ = 10. So I suspect a_n = 2n.

Let’s verify this by induction. Obviously a₁ = 2 = 2(1) so it’s true for 1.

If it’s true for n, then

Hence, by induction, a_n = 2n for all n.

Last edited by JaneFairfax (2007-04-12 14:55:55)

2.

Actually, I cheated slightly – I assumed beforehand that the series was absolutely convergent so I could take liberties with rearranging the summation terms.

Argh! I actually arrived at more or less the same final process as you, only I did it by a tedious route involving more work than necessary. Argh!

Last edited by JaneFairfax (2007-04-12 18:22:41)

4.

Hmm …

Could it be

Let’s see.

so they’re fine.

Yes!

I can write the result in one single formula:

I am gonna give the answer to the last question.
 
   

Last edited by Stanley_Marsh (2007-04-15 16:02:15)