... so that is...
End of 'part 1'... to be continued...
The rth term of a series (for r=1, 2, ...) is . Prove that the sum S of the first n terms of the series is given by
Hence prove that the error in the approximate formula
I have found S correctly, my proof of the second part is not quite satisfactory to me, and I don't know how to do the last part.
For all r, is positive, hence is positive.
When , , hence
For all r,
For this last part, I did find the general term, but don't know how to find the sum of the series.
Thank you! it doesn't look very encouraging trying to simplify the result, but I'm supposed to prove that it is less than S/4. I thought it would have something to do with finding the sum of the series, which I don't know how, then finding the difference.
I think that the 'spirit' of the problem is the following. Let suppose we have 'forgotten' that is...
... and we have to find some pratical method to compute the sum. The general term tend to 0 like and that means that we have to sum a very high number of terms. If however we consider that is...
... with the first series on the right which we know is equal to 2 and the second which has its general term that tends to 0 like , our task becomes much less difficult ...