2
$\begingroup$

Let $ f : [1,\infty) \to (0,\infty) $ be a twice differentiable decreasing function such that $f''(x)$ is positive for $x \in (1, \infty $). For each positive integer $n$, let $ a_{n} $ denote the area of the region bounded by the graph of $f$ and the line segment joining the points $(n, f(n))$ and $(n + 1, f(n + 1))$. I want to show
a. $\displaystyle \sum_{n=1}^ \infty a_{n}<\frac 1 2 (f(1)-f(2))$

b. $\displaystyle\lim_{n \to \infty} \left[ \sum_{k=1}^nf(k)-\frac1 2(f(1)+f(n))- \int_1^n f(x)dx\right]$ exists.

c. $\displaystyle\lim_{n \to \infty} \left[ \sum_{k=1}^nf(k)\int_1^n f(x)dx\right]$ exists.

  • 1
    And, consider registering your account. This has several benefits in addition to helping system maintain your posts efficiently, and helping you edit your own posts.2012-03-10

2 Answers 2