I'm a beginner at summations, and my first instinct for this sum was to use a partial fraction. This didn't really work even after I tried factoring the polynomials, i think because the numerator has a higher exponent. If you could help me figure out how to work through the question, or point me in the direction of how to begin I'd appreciate it thanks!
Evaluate $\sum\limits_{k=1}^n (k^{3} +k^{2} +1) / (k^{2} +k)$
discrete mathematicssummation
Best Answer
Note that the summation is equivalent to $\sum_{k=1}^n k + \dfrac{1}{k^2+k} = \sum_{k=1}^n k + \dfrac{1}{k} - \dfrac{1}{k+1}$. The first part ($\sum_{k=1}^n k$) is equal to $\dfrac{k^2 + k}{2}$, and the second part, is, by telescoping, equal to $1 - \frac{1}{k+1}$, so we have $\dfrac{k(k+1)}{2} + 1 - \dfrac{1}{k+1}$.