Question: Given a summation $\sum_{k=1}^n a_k$ , where are starting value $k=1$, and our ending value is n. Come up with a summation example where the sum is less when $n=10$ and $n=5$.
My example is $1/e_n$. I get around 0.03368 when $n=5$ and 0.000453 when $n=10$.
My Question: Are there any other examples without the use of exponentials? If so is their a trick to coming up with these examples, I mean like this was suppose to be an easy exercise, but I kept thinking about it and realized how weak my math skills are. Generally are there any math cues to make summations smaller like there are for making a fraction smaller ( i.e make the denominator bigger than the numerator)
The inequality $$\sum_{k=1}^{10}a_k < \sum_{k=1}^{5}a_k$$ is equivalent to the inequality $$\sum_{k=6}^{10}a_k < 0$$ so no need for a special trick; just make sure the sum of the last $5$ terms is negative.
As regards your example, since all your terms are positive, the sum is increasing, so that example doesn't work.