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Determine formulas for given sums

Consider the formula By differentiating, determine formulas for the following:

1. .
2. .

1. We observe that Thus, using the given formula we have, 2. (Note: There’s an alternate solution to this in the comments that is more direct than this one. Definitely worth taking a look at that alternative.)
Taking derivatives of both sides of the equation we derived in part (a), Then, looking at the second term in the sum on the left, we have, Thus, plugging this back into the expression above, 3. ( Note: This is a lot of algebra. I included as many steps as I thought appropriate, but that still has a lot going on in each step. If there is something that isn’t clear please leave a comment. Also, given all of these equations it seems likely I’ve made at least one typo / mathematical error. If you find one, please point it out.)

1. tasos says:

For the part b, This last step uses the result of part (a).

This provides a direct solution. Hopefully there’s no typo.

• RoRi says:

Hi, thanks! I retyped your comment into latex so that it’s easier to read. Let me know if I made a mistake. This is a nice solution, that I think works also. I’ll leave the solution I have up there and put a note to look at the comments for this alternative. (I’m not anxious to actually take that derivative since I’ll definitely make an algebra mistake.)