Will the original even converge? If not I've wasted a lot of time today
time is never wasted... you get more out of a question that took 10 hours of work and you have failed to solve it than spending 5 minutes in a question which you managed. (Apart from a feel good factor)
time is never wasted... you get more out of a question that took 10 hours of work and you have failed to solve it than spending 5 minutes in a question which you managed. (Apart from a feel good factor)
if you say so..it got to a point where I couldn't think of another way to rewrite it in the hope of being able to cancel. Oh well - I didnt read further than the second line of the solution so i can go again tomorrow.
if you say so..it got to a point where I couldn't think of another way to rewrite it in the hope of being able to cancel. Oh well - I didnt read further than the second line of the solution so i can go again tomorrow.
Interestingly you can use the same trick as in the extra question to prove that each natural number has a binary representation and it is unique!
Interestingly I do not do pure maths, and I do not quite understand your statement. Nevertheless I hope someone else finds this bit of information useful in the study of the subject.
Interestingly I do not do pure maths, and I do not quite understand your statement. Nevertheless I hope someone else finds this bit of information useful in the study of the subject.
Ah fair enough, got a bit over-excited when I realised the same trick was involved; I remember being impressed that you could prove number facts using algebra. {For clarification, the statement means that every natural number can be written uniquely as the sum of powers of 2}.
Ah fair enough, got a bit over-excited when I realised the same trick was involved; I remember being impressed that you could prove number facts using algebra. {For clarification, the statement means that every natural number can be written uniquely as the sum of powers of 2}.
many thanks.
(I don't think we are getting notifications at present.)
Normal was interesting, I think i got there in the end...I wouldn't even know where to start with the DE though....
I hope you got it right I think the series is 6.
it is not a differential equation as such, since it is satisfied for all f(x) (it is a differential identity) it is the equivalent statement to dy/dx = 1/(dx/dy) but for second derivatives.
PS: I have made loads of product operator questions but I thought I give them a day off
it is not a differential equation as such, since it is satisfied for all f(x) (it is a differential identity) it is the equivalent statement to dy/dx = 1/(dx/dy) but for second derivatives.
PS: I have made loads of product operator questions but I thought I give them a day off
I got 6, and Wolfram Alpha agrees. Very satisfying - series are my weakest. I'll take another look...
Good call, I've seen more of that product operator symbol in the last 2 days then I have in all my life.
it is not a differential equation as such, since it is satisfied for all f(x) (it is a differential identity) it is the equivalent statement to dy/dx = 1/(dx/dy) but for second derivatives.
PS: I have made loads of product operator questions but I thought I give them a day off
I agree on six, as does my calculator. I have a feeling that I may have overcomplicated a tad (used knowledge way beyond AS).
The differential question looks intriguing. One for tomorrow morning.