the question i have to answer is:
determine whether $\sum_{n=1}^\infty {1\over n(n+1)} $ is convergent and if so find the sum.
I have used the telescoping series test and found that the terms cancel except for the first term which = 1 and the last term which would = ${1\over k+1}$
Therefore i have that the sum is equal to 1 – ${1\over k+1}$
After this I'm not really sure where to go to find the sum of convergence, could someone please give me a hint for this 🙂
Best Answer
The sum of the series is defined to be the limit of the partial sums (if that limit exists). The $k$th partial sum is, as you've found, $1 - \frac{1}{k+1}$. What's the limit of that as $k \to \infty$? THAT's the sum of your series.