Let $f$ be a positive-valued $C^1$ function on the interval $[1,$ $\infty$ ), and suppose $\int_1^\infty|f'(t)|\,dt<\infty$.
Then I need to prove that the convergence of $\sum_{1}^{\infty} f(k)$ is equivalent to the convergence of $\int_{1}^{\infty}f $. However every estimate I tried does not seem to work at all....The problem seems to be overwhelming. Could anyone please help me?
Fits into one sentence: $$\int_k^{k+1}f(t)\,dt-f(k)=\int_k^{k+1}(f(t)-f(k))\,dt=\int_k^{k+1}\int_k^tf'(s)\,dsdt,$$so $$\begin{align}\left|\int_k^{k+1}f(t)\,dt-f(k)\right| &\le\int_k^{k+1}\int_k^t|f'(s)|\,dsdt \\&\le\int_k^{k+1}\int_k^{k+1}|f'(s)|\,dsdt \\&=\int_k^{k+1}|f'(s)|\,ds,\end{align}$$hence $$\sum_{k=1}^\infty\left|\int_k^{k+1}f(t)\,dt-f(k)\right|\le\int_1^\infty|f'(s)|\,ds<\infty.$$