Question

Explain why the sum $\sum _{k=1}^{100}f\left(2+0.1\left(k-1\right)\right)\left(0.1\right)$can't be a left sum for f on$\left[a,b\right]=\left[2,5\right]$

Short answer

Part of the sum denotes$0.25$and the other part indicates$0.5$

Step-by-step solution

Step 1. Given information

Sum is$\sum _{k=1}^{100}f\left(2+0.1\left(k-1\right)\right)\left(0.1\right)$

Step 2. Explanation

Consider the sum $\sum _{k=1}^{100}f\left(2+0.1\left(k-1\right)\right)\left(0.1\right)$

$\sum _{k=1}^{100}f\left(2+0.1\left(k-1\right)\right)\left(0.1\right)=f\left(2+0.1\left(1-1\right)\right)\left(0.1\right)+f\left(2+0.1\left(2-1\right)\right)\left(0.1\right)+f\left(2+0.1\left(3-1\right)\right)\left(0.1\right)+...$

Part of it denotes 0.25 and other part denotes 0.5

So, the sum is incorrect.