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## Telescoping Sums and Products (Example 2)

This time I will provide both the problem and solution in  the continuing series from Dr. Andreescu’s lecture on Telescoping Sums and Products.

### Example 2.  Evaluate

$\displaystyle\sum^{n}_{k=2}k!(k^2+k+1)$

### Example 2.  Solution

We can write

$\displaystyle\sum^{n}_{k=1}k!(k^2+k+1)=\displaystyle\sum^{n}_{k=1}[(k+1)^2-k)]k!$

$=\displaystyle\sum^{n}_{k=1}[(k+1)!(k+1)-k!k]$

$=(n+1)!(n+1)-1$