Question

Prove that if${{\left\{a_k\right\}}_{k=1}}^{\infty }$ is a sequence of positive real numbers, then the sequence ${{\left\{S_n\right\}}_{n=1}}^{\infty }$, where the sequence Sn = a1 + a2 +···+an, is an increasing sequence.

Short answer

Proved.

Step-by-step solution

Step 1. Given

Consider the sequences${{\left\{a_k\right\}}_{k=1}}^{\infty }$.

Step 2. Proof

$$\begin{gathered} \mathrm{The}\mathrm{sequenc}e{S}_n=a_1+a_2+a_3+...+a_n......\left(1\right) \\ \mathrm{Change}\mathrm{n}\mathrm{to}\mathrm{n}+1\mathrm{in}\mathrm{equation}\left(1\right)S_{n+1}=a_1+a_2+a_3+...+a_n+a_{n+1}.....\left(2\right) \\ \mathrm{Subtract}\left(1\right)\mathrm{from}\left(2\right) \\ {S}_{n+1}-S_n=a_1+a_2+a_3+...+a_n+a_{n+1}-(a_1+a_2+a_3+...+a_n)=a. \\ \mathrm{Hence}{S}_{n+1}-S_n=a_{n+1} \\ \mathrm{Since}{S}_{n+1}\ge S_n \\ \mathrm{Hence}\mathrm{the}\mathrm{sequence}\mathrm{is}\mathrm{increasing}\mathrm{sequence}. \end{gathered}$$