Articles of telescopic series

Computing $\sum\limits_{n=1}^q {1\over (kn-1)(kn)(kn+1)}$ as a telescoping series, when $k\geqslant3$?

One knows that $$S_1(q)=\sum_{n=1}^q {1\over (n-1)(n)(n+1)} = \sum_{n=1}^q\frac12\left({\frac{1}{(n-1)n}-\frac{1}{n(n+1)}}\right)$$ and the RHS can be easily telescoped. The same approach works for $$S_2(q)=\sum_{n=1}^q {1\over (2n-1)(2n)(2n+1)}$$ However, for $$S_3(q)=\sum_{n=1}^q {1\over (3n-1)(3n)(3n+1)}$$ it is impossible to telescope using the same method than in the two cases above. So: How should $$S_k(q)=\sum_{n=1}^q {1\over (kn-1)(kn)(kn+1)}$$ where $k\geqslant3$ is an integer, be […]

Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$

Can someone give an explanation using the definition of convergence in partial sum to show how the above infinite sum converges to 1? Thanks

Sum of reciprocals of product of consecutive integers

We know, for instance, that $$\sum_{r=1}^n r^\overline{3}=\sum_{r=1}^nr(r+1)(r+2)=\frac {n(n+1)(n+2)(n+3)}4=\frac {n^\overline{4}}4$$ and, in general, $$\sum_{r=1}^n r^\overline{m}=\frac {n^\overline{m+1}}{m+1}\tag{1}$$ which is analogous to $$\int_0^n x^m dx=\frac {n^{m+1}}{m+1}\tag{1A}$$ but is there is correspondingly neat form (and analogy) for sum of reciprocals, e.g. $\displaystyle\sum_{r=1}^n \frac 1{r(r+1)(r+2)}$ and in general $$\sum_{r=1}^n \frac 1{r^\overline{m}}\tag{2}$$ ? Or, alternatively, can it be shown that $(2)$ […]

Why does $\sum_1^\infty \frac1{n^3}=\frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$?

Apery’s original proof that $$ \zeta(3) \equiv \sum_1^\infty \frac1{n^3} $$ is irrational starts from an alternating series $$ \zeta(3) = \frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}} $$ There must be a way to see that those two series are equal, but I have tried various telescoping and other techniques and I just can’t see it. Help me out: Show (preferably […]

What is the formula for $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\ldots +\frac{1}{n(n+1)}$

How can I find the formula for the following equation? $$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\ldots +\frac{1}{n(n+1)}$$ More importantly, how would you approach finding the formula? I have found that every time, the denominator number seems to go up by $n+2$, but that’s about as far as I have been able to get: $\frac12 + \frac16 + […]

How do telescopic series work in general and in this specific problem?

$$\sum_{n=1}^\infty\frac{1}{n(n+3)(n+6)}$$ I did the partial fraction decomposition and also plugged in the values. I can’t understand how the eleminating thing works, for example in cases like this where you don’t know what to cancel what am I supposed to do? I’m going to write it in simple math. From the partial fraction decomposition I’ve got: […]