Very possibly this has been asked before, but as I couldn't find it:
Let $(a_n)_{n\in\mathbb N}$ be a sequence, and
$$\sum_{n= 0}^\infty a_n $$
an infinite sum.
I'm looking for general methods of transforming this sum into a telescoping sum, i.e. finding a solution to the recurrence formula
$$a_n= r(n) – r(n-1) $$
This can be seen as an inhomogenous recurrence equation:
$$r(n)= r(n-1)+a_n $$
One failed attempt would be this:
Using the methods of discrete math, we can show that the generating function of $r(n)$ has the closed form:
$$ \sum_{i=0}^\infty r(i)x^i = \frac{r(0) – \sum_{n=d}^\infty a_n x^n}{1-x}$$
However, to make use of this, we would first need to find a closed form for
$$ \sum_{n=d}^\infty a_n x^n$$
However, if we find a closed form for that, we might as well find a closed form for
$$\sum_{n=0}^\infty a_n x^n$$
, substitute $x=1$ and be done with it.
TL, DR: In this form, my approach either is missing something, or plain pointless. What are other possible ways of going about solving this recurrence equation, or other general ways to transform sums into telescopic-sum-form?
Best Answer
You are essentially asking for a general method to sum any finite or infinite series (or indefinite sum). Correct? There is no such method. For certain special families of series, there do exist methods. For instance, summing of polynomial sequences, that is, $\sum_n n^k$ using Bernoulli numbers. An important special case similar to telescoping series is using Wilf–Zeilberger pairs which requires a computer algorithm.