Inequality

Let $a_k>0$. Prove that:

$$\sum_{k=1}^n \frac{2k+1}{a_1+a_2+\cdots+a_k}<4\sum_{k=1}^n\frac1{a_k}$$

Solution

Proof of AM - GM inequality using concavity

The AM - GM (arithmetic - geometric mean ) inequality is expressed as follows:

$$\sum_{k=1}^{n}a_k \geq n \sqrt[n]{\prod_{k=1}^{n}a_k} \tag{1}$$

One proof was given by the French mathematician  Augustin Luis Cauchy in its lecture book that he had prepared for his students. The proof is based on induction , the so called "back and forth" form of induction. Since then many proofs of this inequality have been discovered. In this topic we give a proof based on the concavity of the $\log $ function.

Proof

Inequality and sequence

Prove that for any sequence of positive real terms $a_1, \; a_2, \dots, a_n$ the following inequality holds:

$$\frac{1}{\frac{1}{a_1}}+ \frac{2}{\frac{1}{a_1}+ \frac{1}{a_2}}+ \cdots+ \frac{n}{\frac{1}{a_1}+ \frac{1}{a_2}+\cdots+ \frac{1}{a_n}}< 2\sum_{i=1}^{n}a_i$$

Solution: