Thanks very much for the kind help.
May I know how and what is the idea which you employ in your solution making it so neat and simply. Is the f(r) unique and is there a general method to find f(r) ?.
Lastly is it possible to find a g(r) such that the nth term= g(r)-g(r-1) allowing us to find the sum?
If you are told to evaluate an infinite sum you should either change it into a known form of a function or you should make it a telescoping sum. Now to make it a telescoping sum you need to look for the following things:
- an obvious way to apply partial fractions decomposition
- add and subtract something in the numerator so that part of it looks like something that we can cancel on the bottom.
That is usually a good start. There is a lot of intuition involved in some examples and less in others.
Thanks very much for the reply.
Can I know how to find g(r) such that the nth term= g(r)-g(r-1) thus allowing us to find the sum of the above 3 problem.
How to apply telescope method in question 3?
I cannot wrap around my head to get started in problem3
Thanks
3)
The sum can be written as...
(1)
In order to solve the second term of (1) we have to introduce the function digamma, defined as...
(2)
... where is the 'Euler's constant'. From (2) You can derive the following basic identity...
(3)
Now setting in (3) You obtain...
(4)
... so that is...
(5)
... and finally You can write...
(6)
For more information about the digamma function see here...
Digamma function - Wikipedia, the free encyclopedia
Kind regards